Locus of the Midpoint of a Segment from an External Point to a Circle
2013-12-04 07:53
This applet demonstrates the locus formed by the midpoint of a segment connecting a fixed external point and a variable point on a circle. Let A be the external point and P the moving point on the circle. The midpoint Q of AP traces a new circle as P moves. This construction uses points, segments, and the locus command to visually explore plane geometry, serving as an excellent teaching tool.
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="4.2" version="4.2.56.0" id="ed192a63-8f22-4088-83fa-311e657da290" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="922" height="690" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.17787418655097614" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1" size="400" window="324,351,700,550" />
<view id="4" toolbar="0 59 || 2020 , 2021 , 2022 , 66 || 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 | 1011 || 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="true" location="1" size="895" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="false" location="3" size="164" window="100,100,250,400" />
<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="150" window="50,50,500,500" />
<view id="64" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
</views>
<toolbar show="true" items="0 39 59 | 1 501 67 , 5 19 , 72 | 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 66 68 | 25 52 60 61 | 40 41 42 , 27 28 35 , 6" position="1" help="false" />
<input show="true" cmd="true" top="false" />
<dockBar show="true" east="true" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="12"/>
<graphicsSettings javaLatexFonts="false"/>
</gui>
<euclidianView>
<size width="895" height="524"/>
<coordSystem xZero="176.03231659942" yZero="256.6622039351285" scale="63.777902343750014" yscale="63.7779023437501"/>
<evSettings axes="true" grid="true" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="13" gridType="0"/>
<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="10"/>
<axis id="0" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0"/>
<coordStyle val="0"/>
<angleFromInvTrig val="false"/>
</kernel>
<scripting blocked="false" disabled="false"/>
<construction title="AP的中點所形成的圖形" author="" date="4 12 月 2013">
<element type="conic" label="c">
<show object="true" label="false"/>
<objColor r="0" g="0" b="204" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<lineStyle thickness="6" 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="-4.0" A3="0.0" A4="0.0" A5="0.0"/>
<eqnStyle style="specific"/>
</element>
<command name="Point">
<input a0="xAxis"/>
<output a0="A"/>
</command>
<element type="point" label="A">
<show object="true" label="true"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="1"/>
<fixed val="true"/>
<coords x="6.0" y="0.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Point">
<input a0="c"/>
<output a0="P"/>
</command>
<element type="point" label="P">
<show object="true" label="true"/>
<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="true"/>
<coords x="-1.5406844469532057" y="-1.275261320246362" z="1.0"/>
<pointSize val="6"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="P" a1="A"/>
<output a0="a"/>
</command>
<element type="segment" label="a">
<show object="true" label="false"/>
<objColor r="0" g="153" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-1.275261320246362" y="7.540684446953206" z="7.651567921478172"/>
<lineStyle thickness="6" type="10" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Midpoint">
<input a0="a"/>
<output a0="Q"/>
</command>
<element type="point" label="Q">
<show object="true" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="2.229657776523397" y="-0.637630660123181" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Locus">
<input a0="Q" a1="P"/>
<output a0="loc1"/>
</command>
<element type="locus" label="loc1">
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<lineStyle thickness="7" type="15" typeHidden="1"/>
</element>
<expression label="text1" exp=""P為圓\; x^2+y^2=4\;上任意點, 且A點坐標為(6,0), 試求""/>
<element type="text" label="text1">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="2.0" size="12" style="0"/>
<startPoint x="0.455421925385044" y="3.747233635106012" z="1.0"/>
</element>
<expression label="text2" exp=""\overline{AP}""/>
<element type="text" label="text2">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="false" sizeM="2.0" size="12" style="0"/>
<startPoint x="0.5169138869273369" y="3.0521888739133876" z="1.0"/>
</element>
<expression label="text3" exp=""的中點所形成的圖形""/>
<element type="text" label="text3">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="false" sizeM="2.0" size="12" style="0"/>
<startPoint x="1.2695256574005924" y="2.6131653411373237" z="1.0"/>
</element>
<expression label="text4" exp=""解: 設P(a,b), 則 \;a^2+b^2=4""/>
<element type="text" label="text4">
<show object="true" label="false"/>
<condition showObject="b"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.0" size="0" style="0"/>
<startPoint x="3.5442660667023596" y="-0.8607634896818219" z="1.0"/>
</element>
<expression label="text5" exp=""\overline{AP} 中點\;Q(x,y)""/>
<element type="text" label="text5">
<show object="true" label="false"/>
<condition showObject="b"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.0" size="0" style="0"/>
<startPoint x="4.013422737250999" y="-1.291008218192677" z="1.0"/>
</element>
<expression label="text6" exp=""(x,y)=(\frac{a+6}{2},\frac{b+0}{2})""/>
<element type="text" label="text6">
<show object="true" label="false"/>
<condition showObject="b"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.0" size="0" style="0"/>
<startPoint x="3.96638450159642" y="-1.6829935153141629" z="1.0"/>
</element>
<expression label="text7" exp=""a=2x-y, b=2y \\代入\; a^2+b^2=4 \\可得 (2x-6)^2+(2y)^2=4 \\即 (x-3)^2+y^2=1""/>
<element type="text" label="text7">
<show object="true" label="false"/>
<condition showObject="b"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="false" sizeM="1.0" size="0" style="0"/>
<startPoint x="3.982063913481279" y="-2.4042464620176993" z="1.0"/>
</element>
<expression label="text8" exp=""按這個鈕可讓P點動起來!""/>
<element type="text" label="text8">
<show object="true" label="false"/>
<objColor r="32" g="32" b="32" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="false" sizeM="1.4" size="5" style="1"/>
<startPoint x="-2.6973655494688513" y="-3.5802023533821568" z="1.0"/>
</element>
<element type="boolean" label="b">
<value val="false"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="501" y="74"/>
<labelMode val="0"/>
<caption val="看計算過程"/>
</element>
</construction>
</geogebra>