DSE 圆：圆心角与圆周角定理演示
2016-12-19 06:42
本 GeoGebra 范例演示了圆中圆心角与圆周角的几何关系。通过构造圆、点、线段和圆弧，动态展示同弧所对的圆心角是圆周角的两倍这一经典定理。用户可通过拖动圆上的点，实时观察角度和距离的变化，深入理解圆的几何性质，适用于中学数学的几何教学与学习。
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.300.0" id="ca0e1d97-ccf2-4fa3-9b56-57c3217795d4" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="900" height="687" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.0" orientation="1" />
</panes>
<views>
<view id="4" toolbar="0 || 2020 , 2021 , 2022 , 66 || 2001 , 2003 , 2002 , 2004 , 2005 || 2040 , 2041 , 2042 , 2044 , 2043" visible="false" inframe="false" stylebar="false" location="1,1" size="200" window="0,0,400,400" />
<view id="1" visible="true" inframe="false" stylebar="true" location="1" size="884" window="0,0,400,400" />
<view id="2" visible="false" inframe="false" stylebar="false" location="3" size="200" window="0,0,200,400" />
<view id="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 || 1008 1009 || 66 68 || 6" visible="false" inframe="false" stylebar="false" location="1" size="150" window="50,50,500,500" />
<view id="16" visible="false" inframe="false" stylebar="false" location="1" size="150" window="50,50,500,500" />
<view id="32" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="64" toolbar="0" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="4097" visible="false" inframe="true" stylebar="true" location="1" size="150" window="284,247,500,500" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="512" toolbar="0 | 1 501 5 19 , 67 | 2 15 45 18 , 7 37 | 514 3 9 , 13 44 , 47 | 16 | 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 | 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" size="300" window="50,50,500,500" />
</views>
<toolbar show="true" items="0 39 | 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 | 30 29 54 32 31 33 | 17 26 62 73 , 14 68 | 25 52 60 61 | 40 41 42 , 27 28 35 , 6" position="1" help="false" />
<input show="true" cmd="true" top="algebra" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="18"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="884" height="483"/>
<coordSystem xZero="278.2881869671948" yZero="288.9021736660397" scale="33.58853558079322" yscale="33.58853558079327"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="1" rightAngleStyle="1" checkboxSize="13" gridType="0"/>
<bgColor r="255" g="255" b="255"/>
<axesColor r="64" g="64" b="64"/>
<gridColor r="192" g="192" b="192"/>
<lineStyle axes="1" grid="10"/>
<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="1"/>
<angleUnit val="degree"/>
<algebraStyle val="0" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<element type="point" label="O_1">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-9" y="-2"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<coords x="NaN" y="NaN" z="NaN"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Circle">
<input a0="O_1" a1="4"/>
<output a0="c"/>
</command>
<element type="conic" label="c">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eigenvectors x0="1.0" y0="0.0" z0="1.0" x1="-0.0" y1="1.0" z1="1.0"/>
<matrix A0="1.0" A1="1.0" A2="44.07729634581489" A3="0.0" A4="-7.374905096530612" A5="-2.384967748834707"/>
<eqnStyle style="specific"/>
</element>
<command name="Point">
<input a0="c"/>
<output a0="A_1"/>
</command>
<element type="point" label="A_1">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-17" y="26"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<coords x="NaN" y="NaN" z="NaN"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Point">
<input a0="c"/>
<output a0="B_1"/>
</command>
<element type="point" label="B_1">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="1" y="22"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<coords x="NaN" y="NaN" z="NaN"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="B_1" a1="O_1"/>
<output a0="d"/>
</command>
<element type="segment" label="d">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="NaN" y="NaN" z="NaN"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="O_1" a1="A_1"/>
<output a0="e"/>
</command>
<element type="segment" label="e">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="NaN" y="NaN" z="NaN"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Angle">
<input a0="A_1" a1="O_1" a2="B_1"/>
<output a0="χ"/>
</command>
<element type="angle" label="χ">
<angleStyle val="0"/>
<value val="1.8501676009711232"/>
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0.1"/>
<layer val="0"/>
<labelOffset x="-11" y="27"/>
<labelMode val="2"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="30"/>
</element>
<command name="Distance">
<input a0="A_1" a1="B_1"/>
<output a0="距离AB"/>
</command>
<element type="numeric" label="距离AB">
<value val="NaN"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<command name="Distance">
<input a0="A_1" a1="B_1"/>
<output a0="a"/>
</command>
<element type="numeric" label="a">
<value val="NaN"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<command name="Circumference">
<input a0="c"/>
<output a0="圆周长c"/>
</command>
<element type="numeric" label="圆周长c">
<value val="25.132741228718345"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<command name="PointIn">
<input a0="c"/>
<output a0="点c_1"/>
</command>
<element type="point" label="点c_1">
<show object="false" label="true"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<coords x="NaN" y="NaN" z="NaN"/>
<coordStyle style="cartesian3d"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Circumference">
<input a0="c"/>
<output a0="b"/>
</command>
<element type="numeric" label="b">
<value val="25.132741228718345"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<command name="PointIn">
<input a0="c"/>
<output a0="点c"/>
</command>
<element type="point" label="点c">
<show object="false" label="true"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<coords x="NaN" y="NaN" z="NaN"/>
<coordStyle style="cartesian3d"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="CircleArc">
<input a0="O_1" a1="A_1" a2="B_1"/>
<output a0="h"/>
</command>
<element type="conicpart" label="h">
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eigenvectors x0="1.0" y0="0.0" z0="1.0" x1="-0.0" y1="1.0" z1="1.0"/>
<matrix A0="1.0" A1="1.0" A2="44.07729634581489" A3="0.0" A4="-7.374905096530612" A5="-2.384967748834707"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Length">
<input a0="h"/>
<output a0="i"/>
</command>
<element type="numeric" label="i">
<value val="NaN"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<command name="PointIn">
<input a0="h"/>
<output a0="点h"/>
</command>
<element type="point" label="点h">
<show object="false" label="true"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<coords x="NaN" y="NaN" z="NaN"/>
<coordStyle style="cartesian3d"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<element type="point" label="O">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="0.19402508297049814" y="1.1068100354199015" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<element type="point" label="D_1">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="4.290678951385464" y="0.25821744839108907" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Circle">
<input a0="O" a1="D_1"/>
<output a0="f"/>
</command>
<element type="conic" label="f">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eigenvectors x0="1.0" y0="0.0" z0="1.0" x1="-0.0" y1="1.0" z1="1.0"/>
<matrix A0="1.0" A1="1.0" A2="-16.24000810903165" A3="0.0" A4="-0.19402508297049814" A5="-1.1068100354199015"/>
<eqnStyle style="specific"/>
</element>
<command name="Point">
<input a0="f"/>
<output a0="C"/>
</command>
<element type="point" label="C">
<show object="true" label="true" ev="4"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="-0.8170959793504236" y="5.166405642582095" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Point">
<input a0="f"/>
<output a0="B"/>
</command>
<element type="point" label="B">
<show object="true" label="true" ev="4"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="4.3681995534949385" y="0.8258299898978635" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Point">
<input a0="f"/>
<output a0="A"/>
</command>
<element type="point" label="A">
<show object="true" label="true" ev="4"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-2.685879035588642" y="-1.9277961275385906" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="A" a1="C"/>
<output a0="g"/>
</command>
<element type="segment" label="g">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-7.094201770120685" y="1.8687830562382186" z="-15.451535069577215"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="C" a1="B"/>
<output a0="j"/>
</command>
<element type="segment" label="j">
<show object="true" label="false"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="4.340575652684231" y="5.1852955328453625" z="-23.242673185473386"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="point" label="D">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="0.9441260979218998" y="2.5277128698218325" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="A" a1="D"/>
<output a0="k"/>
</command>
<element type="segment" label="k">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-4.455508997360424" y="3.630005133510542" z="-4.969048369560103"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="D" a1="B"/>
<output a0="l"/>
</command>
<element type="segment" label="l">
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="1.7018828799239691" y="3.4240734555730388" z="-10.261866583409986"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Angle">
<input a0="A" a1="D" a2="B"/>
<output a0="α"/>
</command>
<element type="angle" label="α">
<angleStyle val="1"/>
<value val="1.7931779329773618"/>
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelOffset x="0" y="16"/>
<labelMode val="1"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="30"/>
</element>
<command name="Angle">
<input a0="A" a1="C" a2="B"/>
<output a0="β"/>
</command>
<element type="angle" label="β">
<angleStyle val="1"/>
<value val="1.1314157851130984"/>
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="30"/>
</element>
<command name="CircleArc">
<input a0="O" a1="A" a2="B"/>
<output a0="p"/>
</command>
<element type="conicpart" label="p">
<show object="true" label="false"/>
<objColor r="220" g="20" b="60" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eigenvectors x0="1.0" y0="0.0" z0="1.0" x1="-0.0" y1="1.0" z1="1.0"/>
<matrix A0="1.0" A1="1.0" A2="-16.24000810903165" A3="0.0" A4="-0.19402508297049814" A5="-1.1068100354199015"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Length">
<input a0="p"/>
<output a0="m"/>
</command>
<element type="numeric" label="m">
<value val="9.46682903429168"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<command name="PointIn">
<input a0="p"/>
<output a0="點p"/>
</command>
<element type="point" label="點p">
<show object="false" label="true"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<coords x="1.101141296690955" y="-2.9605820196492414" z="1.0"/>
<coordStyle style="cartesian3d"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<expression label="文字p" exp=""arc length of arc AB = " + m + """ />
<element type="text" label="文字p">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<absoluteScreenLocation x="291" y="26"/>
</element>
<command name="Distance">
<input a0="A" a1="O"/>
<output a0="distanceAO"/>
</command>
<element type="numeric" label="distanceAO">
<value val="4.1836207161213315"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="文字AO" exp=""" + (Name[O] + (Name[A])) + " = " + distanceAO + """ />
<element type="text" label="文字AO">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<absoluteScreenLocation x="424" y="60"/>
</element>
<command name="Distance">
<input a0="A" a1="B"/>
<output a0="distanceAB"/>
</command>
<element type="numeric" label="distanceAB">
<value val="7.572481861027834"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="文字AB" exp="Name[A] + (Name[B]) + " = " + distanceAB" />
<element type="text" label="文字AB">
<show object="true" label="false" ev="20"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<startPoint x="4.397685414938738" y="5.921728060683202" z="1.0"/>
</element>
</construction>
</geogebra>