直线与圆相切的几何构造
2020-12-30 11:49
本 GeoGebra 范例演示直线与圆相切的几何构造过程。通过 Circle、Tangent、Intersect、Distance、Segment、Center 等命令，动态展示如何从给定点向圆作切线，并验证切线与过切点的半径互相垂直这一核心性质。构造包含圆心、切点、切线段及距离标注等关键元素，支持交互操作，帮助学习者直观理解圆的切线定义、切线长定理及切线与半径的垂直关系，适用于平面几何与解析几何的教学场景。
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.620.0" app="classic" platform="w" id="93b1039e-817f-4ddf-9a12-4d97954fdf52" xsi:noNamespaceSchemaLocation="https://ggb123.cn/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1334" height="536" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.13043478260869565" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="false" stylebar="true" location="1,1,1,1" size="400" window="469,20,428,687" />
<view id="4" toolbar="0 || 2020 , 2021 , 2022 , 66 || 2001 , 2003 , 2002 , 2004 , 2005 || 2040 , 2041 , 2042 , 2044 , 2043" visible="false" inframe="false" stylebar="true" location="1,1" size="300" window="100,100,600,400" />
<view id="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 || 1008 1009 || 66 68 || 6" visible="false" inframe="false" stylebar="false" location="1,3" size="300" window="100,100,600,400" />
<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" size="800" window="100,100,600,400" />
<view id="1" visible="true" inframe="false" stylebar="true" location="1" size="1150" window="100,100,600,400" />
<view id="2" visible="true" inframe="false" stylebar="true" location="3" size="174" tab="ALGEBRA" window="100,100,600,400" />
<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="712" window="50,50,500,500" />
<view id="64" toolbar="0" visible="false" inframe="false" stylebar="true" location="1" size="150" 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="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
</views>
<toolbar show="false" items="0 39 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 || 100001 100002 100003 100004 100005 100006 100007 100008 100009 100010 100011 100012 100013 100014 100015 " position="1" help="false" />
<input show="false" cmd="true" top="true" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="12"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="1150" height="534"/>
<coordSystem xZero="508.3736115761818" yZero="226.94188559736827" scale="47.53062404941087" yscale="47.53061564483533"/>
<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"/>
<labelStyle axes="0" serif="true"/>
<axis id="0" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<algebraView>
<mode val="1"/>
<auxiliary show="true"/>
</algebraView>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<tableview min="-2" max="2" step="1"/>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<element type="numeric" label="r">
<value val="2.9"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="5" absoluteScreenLocation="true" width="200" x="14" y="76" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation speed="1" type="0" playing="false"/>
</element>
<expression label="text1" exp=""直线与圆相切""/>
<element type="text" label="text1">
<show object="true" label="false" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="0" b="0" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="0"/>
<absoluteScreenLocation x="7" y="9"/>
</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="-39" y="15"/>
<labelMode val="3"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="-1.7034261525334793" y="0.9079941082820135" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<caption val="$\huge{C}$"/>
</element>
<command name="Circle">
<input a0="C" a1="r"/>
<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"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" 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="-4.683886042190139" A3="0" A4="1.7034261525334793" A5="-0.9079941082820135"/>
<eqnStyle style="specific"/>
</element>
<element type="point" label="P">
<show object="true" label="true" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="11" y="11"/>
<labelMode val="3"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="4.452421836578879" y="0.9034573824636396" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<caption val="$\huge{P}$"/>
</element>
<command name="Tangent">
<input a0="P" a1="c"/>
<output a0="f" a1="g"/>
</command>
<element type="line" 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="-2.5615658604543956" y="-4.787782930490307" z="15.730729607106984"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<element type="line" label="g">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="2.5545060983603483" y="-4.791553364033381" z="-7.044774473809085"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="c" a1="f"/>
<output a0="A"/>
</command>
<element type="point" label="A">
<show object="true" label="true" ev="4"/>
<condition showObject="a ≠ r"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="3"/>
<coords x="-0.33536109391142765" y="3.465023242918035" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
<caption val="$\huge{A}$"/>
</element>
<command name="Intersect">
<input a0="c" a1="g"/>
<output a0="B"/>
</command>
<element type="point" label="B">
<show object="true" label="true" ev="4"/>
<condition showObject="a ≠ r"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelOffset x="-8" y="36"/>
<labelMode val="3"/>
<coords x="-0.33913152745450165" y="-1.6510487158967089" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
<caption val="$\huge{B}$"/>
</element>
<command name="Distance">
<input a0="C" a1="P"/>
<output a0="a"/>
</command>
<element type="numeric" label="a">
<value val="6.1558496608461635"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<element type="button" label="button1">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="22" y="104"/>
<labelMode val="0"/>
<fixed val="true"/>
<auxiliary val="true"/>
<ggbscript val="SetValue(P,C+(r,0))"/>
<font serif="false" sizeM="2" size="12" style="0"/>
<caption val="相切"/>
</element>
<command name="Segment">
<input a0="C" a1="A"/>
<output a0="h"/>
</command>
<element type="segment" label="h">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-2.5570291346360214" y="1.3680650586220517" z="-5.59790531370436"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="C" a1="P"/>
<output a0="i"/>
</command>
<element type="segment" label="i">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.004536725818373943" y="6.155847989112359" z="-5.58174572818781"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Circle">
<input a0="A" a1="C" a2="P"/>
<output a0="d"/>
</command>
<element type="conic" label="d">
<show object="true" label="false" ev="4"/>
<condition showObject="b ∧ a > r"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" 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="-6.764037818178734" A3="0" A4="-1.3744978420226996" A5="-0.9057257453728261"/>
<eqnStyle style="specific"/>
</element>
<element type="boolean" label="b">
<value val="false"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="25" y="163"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<caption val="$\large{过圆外一点作圆的切线}$"/>
</element>
<command name="Center">
<input a0="d"/>
<output a0="D"/>
</command>
<element type="point" label="D">
<show object="true" label="true" ev="4"/>
<condition showObject="b ≟ true"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="1.3744978420226996" y="0.9057257453728261" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
</construction>
</geogebra>