三角形的垂心几何构造与性质演示(一)
2020-10-13 09:58
本GeoGebra范例详细演示了三角形垂心的几何构造过程及其基本性质。通过作三角形各顶点到对边的垂线(高线),直观展示三条高线交于一点(垂心H)的几何定理。范例综合使用了线段、垂线、交点、角、多边形等基础几何作图命令,帮助学习者深入理解三角形垂心的位置特征及其在锐角、直角、钝角三角形中的变化规律,适合中学平面几何教学与探究。
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.609.0" app="classic" platform="w" id="4DED8B2A-0123-4D8A-8B40-F02B392E2590" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1360" height="610" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="1" orientation="1" />
</panes>
<views>
<view id="512" toolbar="0 | 1 501 5 19 , 67 | 2 15 45 18 , 7 37 | 514 3 9 , 13 44 , 47 | 16 51 | 551 550 11 , 20 22 21 23 , 55 56 57 , 12 | 69 | 510 511 , 512 513 | 533 531 , 534 532 , 522 523 , 537 536 , 535 , 538 | 521 520 | 36 , 38 49 560 | 571 30 29 570 31 33 | 17 | 540 40 41 42 , 27 28 35 , 6 , 502" visible="false" inframe="false" stylebar="false" location="1,1,1" size="500" window="100,100,600,400" />
<view id="4" toolbar="0 || 2020 , 2021 , 2022 || 2001 , 2003 , 2002 , 2004 , 2005 || 2040 , 2041 , 2042 , 2044 , 2043" visible="false" inframe="false" stylebar="false" location="1,1" size="300" window="100,100,600,400" />
<view id="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 || 1008 | 1009 || 6" visible="false" inframe="false" stylebar="false" location="1,3" size="300" window="100,100,600,400" />
<view id="1" visible="true" inframe="false" stylebar="false" location="3" size="1360" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="true" location="3" size="380" tab="ALGEBRA" window="100,100,600,400" />
<view id="16" visible="false" inframe="false" stylebar="false" location="1" size="300" window="50,50,500,500" />
<view id="32" visible="false" inframe="false" stylebar="true" location="1" size="300" window="50,50,500,500" />
<view id="64" toolbar="0" visible="false" inframe="false" stylebar="false" location="1" size="480" window="50,50,500,500" />
<view id="128" visible="false" inframe="false" stylebar="false" location="1" size="480" window="50,50,500,500" />
<view id="4097" visible="false" inframe="false" stylebar="true" location="1" size="288" window="100,100,700,550" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="false" stylebar="true" location="1" size="900" window="50,50,500,500" />
</views>
<toolbar show="false" items="0 73 62 | 1 501 67 , 5 19 , 72 75 76 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 | 16 51 64 , 70 | 10 34 53 11 , 24 20 22 , 21 23 | 55 56 57 , 12 | 36 46 , 38 49 50 , 71 14 68 | 30 29 54 32 31 33 | 25 17 26 60 52 61 | 40 41 42 , 27 28 35 , 6" position="1" help="false" />
<input show="false" cmd="true" top="algebra" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="16"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="1360" height="610"/>
<coordSystem xZero="318" yZero="327.6666666666665" scale="33.33333333333332" yscale="33.33333333333338"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="3"/>
<bgColor r="255" g="255" b="255"/>
<axesColor r="0" g="0" b="0"/>
<gridColor r="192" g="192" b="192"/>
<lineStyle axes="1" grid="0"/>
<axis id="0" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="3" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<tableview min="-2" max="2" step="1"/>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<element type="point" label="A">
<show object="true" label="true" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-4.02" y="6.98" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<element type="point" label="C">
<show object="true" label="true" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="-21" y="25"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-7.23" y="-0.25" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<element type="point" label="B">
<show object="true" label="true" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="-9" y="34"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="5.52" y="-0.13" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="A" a1="C"/>
<output a0="f"/>
</command>
<element type="segment" label="f">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="7.23" y="-3.210000000000001" z="51.47040000000001"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="C" a1="B"/>
<output a0="g"/>
</command>
<element type="segment" label="g">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.12" y="12.75" z="2.3199"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="B" a1="A"/>
<output a0="h"/>
</command>
<element type="segment" label="h">
<show object="true" label="false" ev="4"/>
<condition showObject="u"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-7.11" y="-9.54" z="38.007000000000005"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="OrthogonalLine">
<input a0="A" a1="g"/>
<output a0="i"/>
</command>
<element type="line" label="i">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-12.75" y="-0.12" z="-50.417399999999994"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="OrthogonalLine">
<input a0="C" a1="h"/>
<output a0="j"/>
</command>
<element type="line" label="j">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="9.54" y="-7.11" z="67.19669999999999"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="OrthogonalLine">
<input a0="B" a1="f"/>
<output a0="k"/>
</command>
<element type="line" label="k">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="3.210000000000001" y="7.23" z="-16.779300000000003"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<expression label="text1" exp=""三角形垂心,指的是三角形的三条高与对边或其延长线相交于一点的这个点。\\锐角三角形的垂心在三角形内,直角三角形的垂心在直角顶点上,钝角三角形的垂心在三角形外。""/>
<element type="text" label="text1">
<show object="true" label="false" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="0.7" size="-5" style="1"/>
<absoluteScreenLocation x="422" y="53"/>
</element>
<expression label="text2" exp=""三角形的垂心""/>
<element type="text" label="text2">
<show object="true" label="false" ev="40"/>
<objColor r="255" g="85" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<font serif="false" sizeM="1.4" size="7" style="1"/>
<absoluteScreenLocation x="577" y="30"/>
</element>
<command name="AngularBisector">
<input a0="A" a1="C" a2="B"/>
<output a0="p"/>
</command>
<element type="line" label="p">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.5490141514282407" y="0.8358130541763085" z="-3.7604190512821036"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="AngularBisector">
<input a0="C" a1="B" a2="A"/>
<output a0="q"/>
</command>
<element type="line" label="q">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.3103222094920007" y="-0.950631435571117" z="1.5893965097715985"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="LineBisector">
<input a0="g"/>
<output a0="r"/>
</command>
<element type="line" label="r">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-12.75" y="-0.12" z="-10.924050000000005"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="LineBisector">
<input a0="f"/>
<output a0="s"/>
</command>
<element type="line" label="s">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="3.210000000000001" y="7.23" z="-6.272699999999997"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="Midpoint">
<input a0="g"/>
<output a0="D_{1}"/>
</command>
<element type="point" label="D_{1}">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelOffset x="-6" y="26"/>
<labelMode val="0"/>
<coords x="-0.8550000000000004" y="-0.19" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Midpoint">
<input a0="h"/>
<output a0="E_{1}"/>
</command>
<element type="point" label="E_{1}">
<show object="false" label="false"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.75" y="3.4250000000000003" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="C" a1="E_{1}"/>
<output a0="c"/>
</command>
<element type="segment" label="c">
<show object="false" label="false"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-3.6750000000000003" y="7.98" z="-24.575250000000004"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="A" a1="D_{1}"/>
<output a0="d"/>
</command>
<element type="segment" label="d">
<show object="false" label="false"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="7.170000000000001" y="3.164999999999999" z="6.731700000000004"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="OrthogonalLine">
<input a0="A" a1="g"/>
<output a0="l"/>
</command>
<element type="line" label="l">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-12.75" y="-0.12" z="-50.417399999999994"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="OrthogonalLine">
<input a0="C" a1="h"/>
<output a0="m"/>
</command>
<element type="line" label="m">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="9.54" y="-7.11" z="67.19669999999999"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="OrthogonalLine">
<input a0="B" a1="f"/>
<output a0="n"/>
</command>
<element type="line" label="n">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="3.210000000000001" y="7.23" z="-16.779300000000003"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="l" a1="n"/>
<output a0="H"/>
</command>
<element type="point" label="H">
<show object="true" label="true" ev="4"/>
<condition showObject="o"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelOffset x="1" y="32"/>
<labelMode val="0"/>
<coords x="-366.53131799999994" y="375.7759290000001" z="91.7973"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Intersect">
<input a0="l" a1="g"/>
<output a0="E"/>
</command>
<element type="point" label="E">
<show object="true" label="true" ev="4"/>
<condition showObject="o"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelOffset x="-29" y="22"/>
<labelMode val="0"/>
<coords x="-642.543462" y="-35.628813" z="162.5769"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Intersect">
<input a0="h" a1="m"/>
<output a0="F"/>
</command>
<element type="point" label="F">
<show object="true" label="true" ev="4"/>
<condition showObject="o"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-370.82674799999984" y="840.355317" z="141.56369999999998"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Intersect">
<input a0="n" a1="f"/>
<output a0="D"/>
</command>
<element type="point" label="D">
<show object="true" label="true" ev="4"/>
<condition showObject="o"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelOffset x="-17" y="-4"/>
<labelMode val="0"/>
<coords x="-318.2694390000001" y="286.53432300000014" z="62.57700000000001"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="C" a1="F"/>
<output a0="t"/>
</command>
<element type="segment" label="t">
<show object="true" label="false" ev="4"/>
<condition showObject="o"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-6.186234479601763" y="4.610495508382447" z="-43.57385141042514"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="B" a1="D"/>
<output a0="a"/>
</command>
<element type="segment" label="a">
<show object="false" label="false" ev="4"/>
<condition showObject="o"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-4.708907953401411" y="-10.606045016539623" z="24.614386050625637"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="E" a1="A"/>
<output a0="b"/>
</command>
<element type="segment" label="b">
<show object="true" label="false" ev="4"/>
<condition showObject="o"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-7.199150525074596" y="-0.06775671082423118" z="-28.46764326924674"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Angle">
<input a0="A" a1="E" a2="C"/>
<output a0="α"/>
</command>
<element type="angle" label="α">
<angleStyle val="0"/>
<value val="1.5707963267948968"/>
<show object="true" label="false" ev="4"/>
<condition showObject="o"/>
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelOffset x="-22" y="31"/>
<labelMode val="1"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="30"/>
</element>
<command name="Angle">
<input a0="C" a1="D" a2="B"/>
<output a0="β"/>
</command>
<element type="angle" label="β">
<angleStyle val="0"/>
<value val="1.5707963267948963"/>
<show object="true" label="false" ev="4"/>
<condition showObject="o"/>
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelOffset x="-10" y="15"/>
<labelMode val="1"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="30"/>
</element>
<command name="Circle">
<input a0="A" a1="D" a2="B"/>
<output a0="e"/>
</command>
<element type="conic" label="e">
<show object="true" label="false"/>
<condition showObject="o"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="20" typeHidden="1" opacity="178"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="-23.0978" A3="0" A4="-0.7499999999999994" A5="-3.4249999999999994"/>
<eqnStyle style="specific"/>
</element>
<command name="Circle">
<input a0="A" a1="F" a2="C"/>
<output a0="c_1"/>
</command>
<element type="conic" label="c_1">
<show object="true" label="false"/>
<condition showObject="o"/>
<objColor r="153" g="51" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="15" typeHidden="1" opacity="178"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="27.319599999999973" A3="0" A4="5.624999999999999" A5="-3.364999999999999"/>
<eqnStyle style="specific"/>
</element>
<command name="Segment">
<input a0="D" a1="E"/>
<output a0="f_1"/>
</command>
<element type="segment" label="f_1">
<show object="true" label="false" ev="4"/>
<condition showObject="o"/>
<objColor r="244" g="0" b="153" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="4.7980584784760065" y="1.133801727363855" z="19.211567666511964"/>
<lineStyle thickness="5" type="30" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Angle">
<input a0="F" a1="C" a2="A"/>
<output a0="γ"/>
</command>
<element type="angle" label="γ">
<angleStyle val="0"/>
<value val="0.22263612950451256"/>
<show object="true" label="true" ev="4"/>
<condition showObject="o"/>
<objColor r="204" g="0" b="0" alpha="0.65"/>
<layer val="0"/>
<labelOffset x="11" y="-27"/>
<labelMode val="3"/>
<lineStyle thickness="3" type="0" typeHidden="2" opacity="178"/>
<arcSize val="30"/>
<caption val="1"/>
</element>
<command name="Angle">
<input a0="A" a1="E" a2="D"/>
<output a0="δ"/>
</command>
<element type="angle" label="δ">
<angleStyle val="0"/>
<value val="0.22263612950451264"/>
<show object="true" label="true" ev="4"/>
<condition showObject="o"/>
<objColor r="204" g="0" b="0" alpha="0.55"/>
<layer val="0"/>
<labelOffset x="-6" y="-23"/>
<labelMode val="3"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="30"/>
<caption val="2"/>
</element>
<command name="Angle">
<input a0="A" a1="B" a2="D"/>
<output a0="ε"/>
</command>
<element type="angle" label="ε">
<angleStyle val="0"/>
<value val="0.2226361295045125"/>
<show object="true" label="true" ev="4"/>
<condition showObject="o"/>
<objColor r="204" g="0" b="0" alpha="0.65"/>
<layer val="0"/>
<labelOffset x="-7" y="-21"/>
<labelMode val="3"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="30"/>
<caption val="3"/>
</element>
<expression label="text4" exp=""∵QDEB四点共圆,AB为直径.\\
∴∠2=∠3(共弦共弧)\\
同理:∠1=∠2 ∴∠1=∠2=∠3\\
∵∠1++∠CHD=90°,∠CHD=∠BHF\\
∴∠3+∠FHB=90°。即CF⊥AB""/>
<element type="text" label="text4">
<show object="true" label="false" ev="40"/>
<condition showObject="o"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="234" b="205" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="1"/>
<absoluteScreenLocation x="700" y="402"/>
</element>
<expression label="text5" exp=""垂心的性质:""/>
<element type="text" label="text5">
<show object="true" label="false" ev="40"/>
<objColor r="153" g="51" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="0.7" size="-5" style="1"/>
<absoluteScreenLocation x="694" y="98"/>
</element>
<expression label="text6" exp=""\\1、证明三线交于一点。""/>
<element type="text" label="text6">
<show object="true" label="false" ev="40"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="0.7" size="-5" style="1"/>
<absoluteScreenLocation x="768" y="126"/>
</element>
<element type="boolean" label="o">
<value val="false"/>
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="722" y="127"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<caption val="1"/>
</element>
<expression label="text3" exp=""2、三角形的垂心与三个顶点构成一个垂心组, \\ 即这四点中以任意三点为三角形的顶点,则另一点为这个三角形的垂心""/>
<element type="text" label="text3">
<show object="true" label="false" ev="40"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="0.7" size="-5" style="1"/>
<startPoint x="13.559999999999995" y="4.549999999999988" z="1"/>
</element>
<command name="OrthogonalLine">
<input a0="A" a1="g"/>
<output a0="g_1"/>
</command>
<element type="line" label="g_1">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-12.75" y="-0.12" z="-50.417399999999994"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="OrthogonalLine">
<input a0="C" a1="h"/>
<output a0="h_1"/>
</command>
<element type="line" label="h_1">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="9.54" y="-7.11" z="67.19669999999999"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="g_1" a1="h_1"/>
<output a0="G"/>
</command>
<element type="point" label="G">
<show object="true" label="true" ev="4"/>
<condition showObject="u"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelOffset x="2" y="32"/>
<labelMode val="0"/>
<coords x="-366.53131799999994" y="375.775929" z="91.7973"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Ray">
<input a0="B" a1="G"/>
<output a0="l_1"/>
</command>
<element type="ray" label="l_1">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-4.223540104120709" y="-9.512833318627017" z="22.0772730433248"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Ray">
<input a0="A" a1="G"/>
<output a0="m_1"/>
</command>
<element type="ray" label="m_1">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="2.886459895879291" y="0.027166681372982282" z="11.413945345451332"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Intersect">
<input a0="f" a1="l_1"/>
<output a0="J"/>
</command>
<element type="point" label="J">
<show object="true" label="true" ev="4"/>
<condition showObject="u"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelOffset x="-26" y="-8"/>
<labelMode val="0"/>
<coords x="-418.7612895739875" y="377.0059826783729" z="82.33534862790083"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Intersect">
<input a0="g" a1="m_1"/>
<output a0="K"/>
</command>
<element type="point" label="K">
<show object="true" label="true" ev="4"/>
<condition showObject="u"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelOffset x="-10" y="38"/>
<labelMode val="0"/>
<coords x="-145.4647791703873" y="-8.065971753904527" z="36.80562367422572"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="A" a1="K"/>
<output a0="i_1"/>
</command>
<element type="segment" label="i_1">
<show object="true" label="false" ev="4"/>
<condition showObject="u"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="7.199150525074596" y="0.06775671082423385" z="28.467643269246718"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="B" a1="J"/>
<output a0="j_1"/>
</command>
<element type="segment" label="j_1">
<show object="true" label="false" ev="4"/>
<condition showObject="u"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-4.708907953401409" y="-10.606045016539621" z="24.614386050625626"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Polygon">
<input a0="C" a1="G" a2="B"/>
<output a0="t1" a1="b_1" a2="c_2" a3="g_2"/>
</command>
<element type="polygon" label="t1">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<condition showObject="u"/>
<objColor r="153" g="51" b="0" alpha="0.45"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="b_1">
<show object="true" label="false" ev="4"/>
<condition showObject="u"/>
<objColor r="153" g="51" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-4.343540104120709" y="3.237166681372983" z="-30.594503282449484"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="c_2">
<show object="true" label="false" ev="4"/>
<condition showObject="u"/>
<objColor r="153" g="51" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="4.223540104120709" y="9.512833318627017" z="-22.0772730433248"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="g_2">
<show object="true" label="false"/>
<objColor r="153" g="51" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="0.12" y="-12.75" z="-2.3199"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Polygon">
<input a0="A" a1="G" a2="B"/>
<output a0="t2" a1="b_2" a2="a_1" a3="g_3"/>
</command>
<element type="polygon" label="t2">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<condition showObject="u"/>
<objColor r="204" g="0" b="0" alpha="0.6"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="b_2">
<show object="true" label="false" ev="4"/>
<condition showObject="u"/>
<objColor r="204" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="2.886459895879291" y="0.027166681372982282" z="11.413945345451332"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="a_1">
<show object="true" label="false" ev="4"/>
<condition showObject="u"/>
<objColor r="204" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="4.223540104120709" y="9.512833318627017" z="-22.0772730433248"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="g_3">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-7.11" y="-9.54" z="38.007000000000005"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Polygon">
<input a0="A" a1="G" a2="C"/>
<output a0="t3" a1="c_3" a2="a_2" a3="g_4"/>
</command>
<element type="polygon" label="t3">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<condition showObject="u"/>
<objColor r="0" g="100" b="0" alpha="0.5"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="c_3">
<show object="true" label="false" ev="4"/>
<condition showObject="u"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="2.886459895879291" y="0.027166681372982282" z="11.413945345451332"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="a_2">
<show object="true" label="false" ev="4"/>
<condition showObject="u"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="4.343540104120709" y="-3.237166681372983" z="30.594503282449484"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="g_4">
<show object="true" label="false"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-7.23" y="3.210000000000001" z="-51.47040000000001"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Angle">
<input a0="C" a1="J" a2="B"/>
<output a0="ζ"/>
</command>
<element type="angle" label="ζ">
<angleStyle val="0"/>
<value val="1.5707963267948968"/>
<show object="true" label="false"/>
<condition showObject="u"/>
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="30"/>
</element>
<command name="Angle">
<input a0="A" a1="K" a2="C"/>
<output a0="η"/>
</command>
<element type="angle" label="η">
<angleStyle val="0"/>
<value val="1.5707963267948963"/>
<show object="true" label="false"/>
<condition showObject="u"/>
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="30"/>
</element>
<command name="Ray">
<input a0="C" a1="G"/>
<output a0="k_1"/>
</command>
<element type="ray" label="k_1">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-4.343540104120709" y="3.237166681372983" z="-30.594503282449484"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Intersect">
<input a0="k_1" a1="h"/>
<output a0="I"/>
</command>
<element type="point" label="I">
<show object="true" label="true" ev="4"/>
<condition showObject="u"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-168.83656725562508" y="382.61184707553167" z="64.45362769787347"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="C" a1="I"/>
<output a0="n_1"/>
</command>
<element type="segment" label="n_1">
<show object="true" label="false" ev="4"/>
<condition showObject="u"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-6.1862344796017625" y="4.6104955083824475" z="-43.573851410425135"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Angle">
<input a0="C" a1="I" a2="B"/>
<output a0="θ"/>
</command>
<element type="angle" label="θ">
<angleStyle val="0"/>
<value val="1.5707963267948966"/>
<show object="true" label="false"/>
<condition showObject="u"/>
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="30"/>
</element>
<element type="boolean" label="u">
<value val="false"/>
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="724" y="170"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<caption val="2"/>
</element>
<expression label="text7" exp=""3、6组四点共圆(3组对角互补,3组同侧角相等)""/>
<element type="text" label="text7">
<show object="true" label="false" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="0.7" size="-5" style="1"/>
<absoluteScreenLocation x="774" y="237"/>
</element>
<expression label="text8" exp=""4、垂心H关于三边的对称点,均在△ABC的外接圆上。""/>
<element type="text" label="text8">
<show object="true" label="false" ev="40"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<isLaTeX val="true"/>
<font serif="false" sizeM="0.7" size="-5" style="1"/>
<absoluteScreenLocation x="775" y="274"/>
</element>
<expression label="text9" exp=""5、△ABC,△ABO,△BCO,△ACO的外接圆是等圆。""/>
<element type="text" label="text9">
<show object="true" label="false" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<isLaTeX val="true"/>
<font serif="false" sizeM="0.7" size="-5" style="1"/>
<absoluteScreenLocation x="776" y="313"/>
</element>
<command name="Segment">
<input a0="B" a1="D"/>
<output a0="p_1"/>
</command>
<element type="segment" label="p_1">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-4.708907953401411" y="-10.606045016539623" z="24.614386050625637"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="E" a1="A"/>
<output a0="q_1"/>
</command>
<element type="segment" label="q_1">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-7.199150525074596" y="-0.06775671082423118" z="-28.46764326924674"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="C" a1="F"/>
<output a0="r_1"/>
</command>
<element type="segment" label="r_1">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-6.186234479601763" y="4.610495508382447" z="-43.57385141042514"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Intersect">
<input a0="n" a1="f"/>
<output a0="M"/>
</command>
<element type="point" label="M">
<show object="true" label="true"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelOffset x="-27" y="1"/>
<labelMode val="0"/>
<coords x="-318.2694390000001" y="286.53432300000014" z="62.57700000000001"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Intersect">
<input a0="l" a1="g"/>
<output a0="N"/>
</command>
<element type="point" label="N">
<show object="true" label="true"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-642.543462" y="-35.628813" z="162.5769"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Intersect">
<input a0="h" a1="m"/>
<output a0="L"/>
</command>
<element type="point" label="L">
<show object="true" label="true"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-370.82674799999984" y="840.355317" z="141.56369999999998"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Intersect">
<input a0="p_1" a1="q_1"/>
<output a0="O"/>
</command>
<element type="point" label="O">
<show object="true" label="true"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelOffset x="8" y="12"/>
<labelMode val="0"/>
<coords x="-303.5968958661703" y="311.25418206590047" z="76.0354544352895"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Circle">
<input a0="A" a1="B" a2="C"/>
<output a0="d_1"/>
</command>
<element type="conic" label="d_1">
<show object="true" label="false" ev="4"/>
<condition showObject="v"/>
<objColor r="255" g="85" b="0" alpha="0.05"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="30" typeHidden="1" opacity="178"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="-40.402299867643165" A3="0" A4="0.8685833406864911" A5="-1.253229947939645"/>
<eqnStyle style="specific"/>
</element>
<command name="Circle">
<input a0="O" a1="C" a2="B"/>
<output a0="e_1"/>
</command>
<element type="conic" label="e_1">
<show object="true" label="false" ev="4"/>
<condition showObject="v"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="15" typeHidden="1" opacity="178"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="-39.35190013235683" A3="0" A4="0.8414166593135096" A5="1.6332299479396453"/>
<eqnStyle style="specific"/>
</element>
<command name="Mirror">
<input a0="O" a1="g"/>
<output a0="O'"/>
</command>
<element type="point" label="O'">
<show object="true" label="true" ev="4"/>
<condition showObject="v"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-297.42433319643516" y="-344.5806015934588" z="76.0354544352895"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="C" a1="O'"/>
<output a0="s_1"/>
</command>
<element type="segment" label="s_1">
<show object="true" label="false" ev="4"/>
<condition showObject="v"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="4.281841154269901" y="3.318346740275483" z="31.787298230440257"/>
<lineStyle thickness="5" type="10" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="O'" a1="B"/>
<output a0="t_1"/>
</command>
<element type="segment" label="t_1">
<show object="true" label="false" ev="4"/>
<condition showObject="v"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-4.401841154269901" y="9.431653259724516" z="25.52427809533404"/>
<lineStyle thickness="5" type="10" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="boolean" label="v">
<value val="false"/>
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="733" y="309"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<caption val="3"/>
</element>
<expression label="text10" exp=""等弦等弧""/>
<element type="text" label="text10">
<show object="true" label="false" ev="40"/>
<condition showObject="v"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="false" sizeM="0.7" size="-5" style="1"/>
<startPoint x="15.630000000000003" y="-1.0600000000000032" z="1"/>
</element>
</construction>
</geogebra>