Geometric Construction of the Butterfly Theorem
2020-02-07 04:08
This applet demonstrates the classic Butterfly Theorem. Using GeoGebra, it constructs a circle, a chord, and its midpoint, showing that if two arbitrary chords are drawn through the midpoint, the lines connecting their endpoints intersect the original chord at two points that are equidistant from the midpoint. The construction involves circles, segments, midpoints, and intersections, providing a c
In collections
Lumei Junior High Math Geometry Models
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.574.0" app="classic" platform="w" id="F9929E7C-B680-4855-83B2-EBDED48755C8" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1520" height="593" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.25" 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 | 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="4097" visible="false" inframe="false" stylebar="true" location="1,1" size="550" window="100,100,700,550" />
<view id="1" visible="true" inframe="false" stylebar="false" location="1" size="1520" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="true" location="3" size="380" 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="1362" 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="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="false" stylebar="true" location="1" size="900" window="50,50,500,500" />
<view id="43" visible="false" inframe="false" stylebar="false" location="1" size="450" window="50,50,500,500" />
</views>
<toolbar show="true" 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="true" cmd="true" top="algebra" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="1"/>
<font size="18"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="1520" height="540"/>
<coordSystem xZero="663.6580154290682" yZero="391.3756853090364" scale="21.670711694179182" yscale="21.670711694179196"/>
<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="蝴蝶定理" author="" date="">
<element type="numeric" label="r">
<value val="9"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<bgColor r="0" g="0" b="0" alpha="63"/>
<layer val="0"/>
<labelOffset x="29" y="22"/>
<labelMode val="1"/>
<slider min="1" max="9" absoluteScreenLocation="true" width="100" x="107" y="155" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation speed="1" type="0" playing="false"/>
</element>
<command name="Point">
<input a0="xAxis"/>
<output a0="A_1"/>
</command>
<element type="point" label="A_1">
<show object="false" label="true"/>
<objColor r="125" g="125" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-3.4279392981008936" y="0" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Circle">
<input a0="A_1" a1="r"/>
<output a0="c"/>
</command>
<element type="conic" label="c">
<show object="true" label="false"/>
<objColor r="127" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="9" type="15" typeHidden="1" opacity="204"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="-69.24923216853555" A3="0" A4="3.4279392981008936" A5="0"/>
<eqnStyle style="specific"/>
</element>
<command name="Point">
<input a0="c"/>
<output a0="E_1"/>
</command>
<element type="point" label="E_1">
<show object="false" label="true"/>
<objColor r="125" g="125" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="3"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-6.128024171487432" y="8.585426120846257" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Point">
<input a0="c"/>
<output a0="B"/>
</command>
<element type="point" label="B">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="3"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="4.799391023802105" y="3.6484292201296036" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<caption val="$%n$"/>
</element>
<command name="Point">
<input a0="c"/>
<output a0="A"/>
</command>
<element type="point" label="A">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-30" y="5"/>
<labelMode val="3"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-11.797089072772086" y="3.310186104907262" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<caption val="$%n$"/>
</element>
<command name="Midpoint">
<input a0="A" a1="B"/>
<output a0="O"/>
</command>
<element type="point" label="O">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-18" y="-5"/>
<labelMode val="3"/>
<coords x="-3.498849024484991" y="3.4793076625184325" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
<caption val="$%n$"/>
</element>
<command name="Line">
<input a0="E_1" a1="O"/>
<output a0="g"/>
</command>
<element type="line" label="g">
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="5.106118458327825" y="2.6291751470024413" z="8.717828351756495"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="Line">
<input a0="A" a1="B"/>
<output a0="f"/>
</command>
<element type="line" label="f">
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.33824311522234174" y="16.59648009657419" z="-58.927721964579696"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="Point">
<input a0="c"/>
<output a0="F_1"/>
</command>
<element type="point" label="F_1">
<show object="false" label="true"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="21" y="-2"/>
<labelMode val="3"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="0.2307355147820549" y="8.222779251176327" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Line">
<input a0="F_1" a1="O"/>
<output a0="h"/>
</command>
<element type="line" label="h">
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="4.743471588657894" y="-3.7295845392670457" z="29.573063006130056"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="c" a1="h" a2="2"/>
<output a0="G"/>
</command>
<element type="point" label="G">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-10.555331330282996" y="-5.495478397699976" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Intersect">
<input a0="c" a1="g"/>
<output a0="H" a1="I"/>
</command>
<element type="point" label="H">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="3"/>
<coords x="-6.128024171487432" y="8.585426120846257" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<element type="point" label="I">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="3"/>
<coords x="1.9921993307342038" y="-7.184851929177776" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="F_1" a1="I"/>
<output a0="j"/>
</command>
<element type="segment" label="j">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="3"/>
<coords x="15.407631180354101" y="1.761463815952149" z="-18.03921582948025"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="E_1" a1="I"/>
<output a0="m"/>
</command>
<element type="segment" label="m">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="3"/>
<coords x="15.770278050024032" y="8.120223502221636" z="26.92506611854165"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Intersect">
<input a0="j" a1="m"/>
<output a0="F_2"/>
</command>
<element type="point" label="F_2">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelOffset x="0" y="28"/>
<labelMode val="3"/>
<coords x="193.90999405012437" y="-699.3349376964365" z="97.33463467165251"/>
<pointSize val="4"/>
<pointStyle val="0"/>
<caption val="$%n$"/>
</element>
<command name="Intersect">
<input a0="m" a1="j"/>
<output a0="R"/>
</command>
<element type="point" label="R">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="193.90999405012437" y="-699.3349376964365" z="97.33463467165251"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Point">
<input a0="c"/>
<output a0="F"/>
</command>
<element type="point" label="F">
<show object="true" label="true"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-31" y="-3"/>
<labelMode val="3"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-7.095259121821307" y="8.21892726032706" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<caption val="$%n$"/>
</element>
<command name="Point">
<input a0="c"/>
<output a0="D"/>
</command>
<element type="point" label="D">
<show object="true" label="true"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-4" y="3"/>
<labelMode val="3"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="0.3770757799836546" y="8.1560934432821" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<caption val="$%n$"/>
</element>
<command name="Line">
<input a0="F" a1="O"/>
<output a0="s"/>
</command>
<element type="line" label="s">
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="4.739619597808628" y="3.5964100973363164" z="4.070196197001753"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="Line">
<input a0="D" a1="O"/>
<output a0="t"/>
</command>
<element type="line" label="t">
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="4.676785780763667" y="-3.8759248044686454" z="29.848902238283244"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="c" a1="t"/>
<output a0="V" a1="C"/>
</command>
<element type="point" label="V">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.3770757799836546" y="8.1560934432821" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<element type="point" label="C">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-33" y="50"/>
<labelMode val="3"/>
<coords x="-10.735852388597749" y="-5.25303781270844" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
<caption val="$%n$"/>
</element>
<command name="Intersect">
<input a0="c" a1="s"/>
<output a0="Z" a1="E_3"/>
</command>
<element type="point" label="Z">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-7.095259121821307" y="8.21892726032706" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<element type="point" label="E_3">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="3.5002294913106766" y="-5.744604183529394" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="D" a1="C"/>
<output a0="b"/>
</command>
<element type="segment" label="b">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="13.40913125599054" y="-11.112928168581403" z="85.5818219441759"/>
<lineStyle thickness="7" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="F" a1="C"/>
<output a0="d"/>
</command>
<element type="segment" label="d">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="13.4719650730355" y="-3.640593266776442" z="125.50885431738524"/>
<lineStyle thickness="7" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="D" a1="E_3"/>
<output a0="e"/>
</command>
<element type="segment" label="e">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="13.900697626811492" y="3.123153711327022" z="-30.71434990726336"/>
<lineStyle thickness="7" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Line">
<input a0="A" a1="B"/>
<output a0="f_1"/>
</command>
<element type="line" label="f_1">
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.33824311522234174" y="16.59648009657419" z="-58.927721964579696"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="f_1" a1="d"/>
<output a0="M"/>
</command>
<element type="point" label="M">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="3"/>
<coords x="-1868.4733348115908" y="751.4197062680652" z="222.35579458856438"/>
<pointSize val="4"/>
<pointStyle val="0"/>
<caption val="$%n$"/>
</element>
<command name="Intersect">
<input a0="e" a1="f_1"/>
<output a0="N"/>
</command>
<element type="point" label="N">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="3"/>
<coords x="325.7097633613878" y="829.5253622611023" z="231.75903673251048"/>
<pointSize val="4"/>
<pointStyle val="0"/>
<caption val="$%n$"/>
</element>
<command name="Segment">
<input a0="M" a1="O"/>
<output a0="MO"/>
</command>
<element type="segment" label="MO">
<show object="true" label="true"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="2"/>
<coords x="-0.09995023377027134" y="4.904230095935131" z="-17.41303612946477"/>
<lineStyle thickness="8" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="O" a1="N"/>
<output a0="NO"/>
</command>
<element type="segment" label="NO">
<show object="true" label="true"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="2"/>
<coords x="-0.09995023377027268" y="4.90423009593513" z="-17.41303612946477"/>
<lineStyle thickness="8" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="A" a1="M"/>
<output a0="i_1"/>
</command>
<element type="segment" label="i_1">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.0691713238408993" y="3.394009952351965" z="-12.05082485282508"/>
<lineStyle thickness="7" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="N" a1="B"/>
<output a0="j_1"/>
</command>
<element type="segment" label="j_1">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.06917132384089841" y="3.394009952351966" z="-12.050824852825078"/>
<lineStyle thickness="7" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="F" a1="E_3"/>
<output a0="k_1"/>
</command>
<element type="segment" label="k_1">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="13.963531443856453" y="10.595488613131983" z="11.991323650905737"/>
<lineStyle thickness="7" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="LaTeX">
<input a0="MO" a1="true" a2="true"/>
<output a0="MO_1"/>
</command>
<element type="text" label="MO_1">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="51" b="102" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="false" sizeM="1" size="0" style="0"/>
<startPoint x="-14.203762535324406" y="11.274090853307901" z="1"/>
</element>
<command name="LaTeX">
<input a0="NO" a1="true" a2="true"/>
<output a0="text2"/>
</command>
<element type="text" label="text2">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="0" g="0" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="false" sizeM="1" size="0" style="0"/>
<startPoint x="-7.444184233109539" y="11.36457417262356" z="1"/>
</element>
<command name="LaTeX">
<input a0="MO" a1="true" a2="true"/>
<output a0="蝴蝶定理"/>
</command>
<element type="text" label="蝴蝶定理">
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="false" sizeM="1" size="0" style="0"/>
<startPoint x="-1.0104244122873347" y="9.541015338329813" z="1"/>
</element>
<expression label="text1" exp=""蝴蝶定理""/>
<element type="text" label="text1">
<show object="true" label="true" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<font serif="false" sizeM="2" size="18" style="1"/>
<startPoint x="-8.981035310304516" y="14.698302583932397" z="1"/>
</element>
<command name="Intersect">
<input a0="e" a1="k_1"/>
<output a0="E"/>
</command>
<element type="point" label="E">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="5" y="29"/>
<labelMode val="3"/>
<coords x="362.8842916662101" y="-595.5685549241555" z="103.67442836736012"/>
<pointSize val="4"/>
<pointStyle val="0"/>
<caption val="$%n$"/>
</element>
<expression label="text3" exp=""
移动D点或者F点,观察线段ON和OM的长度变化。

当r变化时,观察线段ON和OM的长度变化""/>
<element type="text" label="text3">
<show object="true" label="true" 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="1" size="0" style="0"/>
<startPoint x="-22.360461122664233" y="13.79378508989078" z="1"/>
</element>
</construction>
</geogebra>