Reflection and Image Formation of a Concave Mirror
2017-02-04 13:21
This GeoGebra applet demonstrates the principles of reflection and image formation by a concave mirror. By constructing the parabola, focus, tangent lines, normal lines, incident rays, and reflected rays, it visually illustrates the law of reflection on the mirror's surface. Users can observe how light rays emitted from an object point converge to form an image after reflection, aiding in the unde
In collections
圓錐曲線
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.307.0" id="92753cfb-a27c-4349-8e28-fdfe54725cf6" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="969" height="698" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.5190918472652218" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1,1" size="400" window="531,100,700,550" />
<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="1,1,1" size="500" window="100,100,600,400" />
<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="300" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="false" location="3,3" size="181" window="100,100,250,400" />
<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="1" visible="true" inframe="false" stylebar="true" location="3" size="503" window="100,100,600,400" />
<view id="16" visible="true" inframe="false" stylebar="true" location="1" size="446" 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="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 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="true" east="true" />
</perspective>
</perspectives>
<labelingStyle val="0"/>
<font size="24"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="503" height="494"/>
<coordSystem xZero="153.23494538321233" yZero="199.83455177604858" scale="18.851591570523354" yscale="69.04181152353189"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" 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="0"/>
<axis id="0" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<euclidianView>
<viewNumber viewNo="2"/>
<size width="446" height="494"/>
<coordSystem xZero="231.00000000000006" yZero="311.00000000000006" scale="49.99999999999999" yscale="50.00000000000001"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" 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="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="1"/>
<angleUnit val="degree"/>
<algebraStyle val="0" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<element type="numeric" label="c">
<value val="6.0"/>
<symbolic val="true" />
<show object="true" label="true" ev="3"/>
<objColor r="255" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="-15" max="15" absoluteScreenLocation="true" width="200.0" x="30.0" y="50.0" fixed="true" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<expression label="para" exp="y^(2) = ((4 * c) * x)" type="conic" />
<element type="conic" label="para">
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="7" 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="0.0" A1="1.0" A2="0.0" A3="0.0" A4="-12.0" A5="0.0"/>
<eqnStyle style="specific"/>
</element>
<expression label="F" exp="(c, 0)" type="point" />
<element type="point" label="F">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-4" y="-6"/>
<labelMode val="0"/>
<coords x="6.0" y="0.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<expression label ="olen" exp="1"/>
<element type="numeric" label="olen">
<value val="1.0"/>
<symbolic val="true" />
<show object="false" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="od">
<value val="10.0"/>
<symbolic val="true" />
<show object="true" label="true" ev="3"/>
<objColor r="0" g="102" b="51" alpha="0.1"/>
<layer val="0"/>
<labelOffset x="-53" y="7"/>
<labelMode val="1"/>
<slider min="0" max="40" absoluteScreenLocation="true" width="200.0" x="30.0" y="90.0" fixed="true" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<expression label="O_T" exp="(od, olen)" type="point" />
<element type="point" label="O_T">
<show object="false" label="true"/>
<objColor r="255" g="215" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="10.0" y="1.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label="O_B" exp="(od, 0)" type="point" />
<element type="point" label="O_B">
<show object="false" label="true"/>
<objColor r="255" g="215" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="10.0" y="0.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Vector">
<input a0="O_B" a1="O_T"/>
<output a0="o"/>
</command>
<element type="vector" label="o">
<show object="true" label="true"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="8" y="12"/>
<labelMode val="0"/>
<coords x="0.0" y="1.0" z="0.0"/>
<lineStyle thickness="6" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
<startPoint exp="O_B"/>
</element>
<command name="Intersect">
<input a0="para" a1="y = y(O_T)"/>
<output a0="Q" a1=""/>
</command>
<element type="point" label="Q">
<show object="true" label="true"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-26" y="-12"/>
<labelMode val="0"/>
<coords x="0.041666666666666664" y="1.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Vector">
<input a0="O_T" a1="Q"/>
<output a0="iQ"/>
</command>
<element type="vector" label="iQ">
<show object="true" label="false"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-9.958333333333334" y="0.0" z="0.0"/>
<lineStyle thickness="7" type="30" typeHidden="1"/>
<coordStyle style="cartesian"/>
<startPoint exp="O_T"/>
</element>
<command name="Tangent">
<input a0="Q" a1="para"/>
<output a0="t_Q" a1=""/>
</command>
<element type="line" label="t_Q">
<show object="true" label="false"/>
<condition showObject="a"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-12.0" y="1.0" z="-0.5"/>
<lineStyle thickness="2" type="20" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="OrthogonalLine">
<input a0="Q" a1="t_Q"/>
<output a0="n_Q"/>
</command>
<element type="line" label="n_Q">
<show object="true" label="false"/>
<condition showObject="a"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-1.0" y="-12.0" z="12.041666666666666"/>
<lineStyle thickness="2" type="20" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Line">
<input a0="Q" a1="Mirror[O_T, n_Q]"/>
<output a0="rQ"/>
</command>
<element type="line" label="rQ">
<show object="true" label="false"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="1.6482758620689686" y="9.820977011494254" z="-9.889655172413793"/>
<lineStyle thickness="6" type="15" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="If">
<input a0="c > 0" a1="Intersect[para, Line[O_T, F], 2]" a2="Intersect[para, Line[O_T, F], 1]"/>
<output a0="P"/>
</command>
<element type="point" label="P">
<show object="true" label="true" ev="8"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-26" y="37"/>
<labelMode val="0"/>
<coords x="0.09092997035229361" y="-1.4772675074119266" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Vector">
<input a0="O_T" a1="P"/>
<output a0="iP"/>
</command>
<element type="vector" label="iP">
<show object="true" label="false"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-9.909070029647706" y="-2.4772675074119266" z="0.0"/>
<lineStyle thickness="7" type="30" typeHidden="1"/>
<coordStyle style="cartesian"/>
<startPoint exp="O_T"/>
</element>
<command name="Tangent">
<input a0="P" a1="para"/>
<output a0="t_P" a1=""/>
</command>
<element type="line" label="t_P">
<show object="true" label="false"/>
<condition showObject="a"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-12.0" y="-1.4772675074119266" z="-1.0911596442275233"/>
<lineStyle thickness="2" type="20" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="OrthogonalLine">
<input a0="P" a1="t_P"/>
<output a0="n_P"/>
</command>
<element type="line" label="n_P">
<show object="true" label="false"/>
<condition showObject="a"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="1.4772675074119266" y="-12.0" z="-17.86153797959449"/>
<lineStyle thickness="2" type="20" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Line">
<input a0="P" a1="Mirror[O_T, n_P]"/>
<output a0="rP"/>
</command>
<element type="line" label="rP">
<show object="true" label="false"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-2.886579864025407E-15" y="10.214035595969953" z="15.088862905475226"/>
<lineStyle thickness="6" type="15" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<expression label ="f" exp="y = 0" type="line"/>
<element type="line" label="f">
<show object="true" label="false"/>
<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"/>
<coords x="0.0" y="1.0" z="0.0"/>
<lineStyle thickness="2" type="20" typeHidden="1"/>
<eqnStyle style="explicit"/>
</element>
<command name="Intersect">
<input a0="rP" a1="rQ"/>
<output a0="i_T"/>
</command>
<element type="point" label="i_T">
<show object="false" label="true"/>
<objColor r="255" g="215" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="249.20066568716345" y="-24.870608513162633" z="16.835548327150534"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Intersect">
<input a0="OrthogonalLine[i_T, y = 0]" a1="y = 0"/>
<output a0="i_B"/>
</command>
<element type="point" label="i_B">
<show object="false" label="true"/>
<objColor r="255" g="215" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="249.20066568716345" y="-0.0" z="16.835548327150534"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Vector">
<input a0="i_B" a1="i_T"/>
<output a0="i"/>
</command>
<element type="vector" label="i">
<show object="true" label="true" ev="4"/>
<objColor r="127" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="12" y="10"/>
<labelMode val="0"/>
<coords x="0.0" y="-1.4772675074119226" z="0.0"/>
<lineStyle thickness="9" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
<startPoint exp="i_B"/>
</element>
<expression label="il" exp="(-y(i))" />
<element type="numeric" label="il">
<value val="1.4772675074119226"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="id" exp="x(i_B)" />
<element type="numeric" label="id">
<value val="14.80205223166268"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="text1" exp=""\frac{1}{od}+\frac{1}{id}=\frac{1}{c}""/>
<element type="text" label="text1">
<show object="true" label="true" ev="43"/>
<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="-3.7600000000000007" y="3.7800000000000002" z="1.0"/>
</element>
<expression label="text2" exp=""\frac{1}{" + (LaTeX[od]) + "}+\frac{1}{" + (LaTeX[id]) + "}=\frac{1}{" + (LaTeX[c]) + "}"" />
<element type="text" label="text2">
<show object="true" label="true" ev="43"/>
<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="-0.10000000000000031" y="3.8400000000000007" z="1.0"/>
</element>
<expression label="text3" exp=""\frac{i}{o}=\frac{id}{od}""/>
<element type="text" label="text3">
<show object="true" label="true" ev="43"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.0" size="0" style="0"/>
<startPoint x="-3.8400000000000007" y="2.2800000000000002" z="1.0"/>
</element>
<expression label="text4" exp=""\frac{" + (LaTeX[il]) + "}{" + (LaTeX[olen]) + "}=\frac{" + (LaTeX[id]) + "}{" + (LaTeX[od]) + "}"" />
<element type="text" label="text4">
<show object="true" label="true" ev="43"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.0" size="0" style="0"/>
<startPoint x="-1.2800000000000002" y="2.26" z="1.0"/>
</element>
<expression label="C" exp="((2 * c), 0)" type="point" />
<element type="point" label="C">
<show object="true" label="true"/>
<objColor r="255" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="12.0" y="0.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label ="O" exp="(0, 0)" type="point"/>
<element type="point" label="O">
<show object="true" label="true" ev="8"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-31" y="17"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="0.0" y="0.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<expression label="text5" exp=""\frac{c}{od}+ \frac{c}{oi} =\frac{i}{i+o}+\frac{o}{i+o}=1""/>
<element type="text" label="text5">
<show object="true" label="true" ev="43"/>
<condition showObject="proof > 2"/>
<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="-3.880000000000001" y="-1.9" z="1.0"/>
</element>
<command name="Polygon">
<input a0="P" a1="O" a2="F"/>
<output a0="tOPF" a1="f_1" a2="p" a3="o_1"/>
</command>
<element type="polygon" label="tOPF">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false" ev="8"/>
<condition showObject="proof ≟ 1 ∨ proof ≟ 3"/>
<objColor r="255" g="215" b="0" alpha="0.26"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="f_1">
<show object="false" label="false" ev="8"/>
<objColor r="255" g="215" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-1.4772675074119266" y="-0.09092997035229361" z="0.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="p">
<show object="false" label="false" ev="8"/>
<objColor r="255" g="215" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="0.0" y="6.0" z="0.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="o_1">
<show object="false" label="false" ev="8"/>
<objColor r="255" g="215" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="1.4772675074119266" y="-5.909070029647706" z="-8.863605044471559"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Polygon">
<input a0="P" a1="Q" a2="O_T"/>
<output a0="tQPO_T" a1="o_t" a2="p_1" a3="q"/>
</command>
<element type="polygon" label="tQPO_T">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false" ev="8"/>
<condition showObject="proof ≟ 1 ∨ proof ≟ 3"/>
<objColor r="0" g="100" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="o_t">
<show object="false" label="false" ev="8"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-2.4772675074119266" y="-0.049263303685626945" z="0.15248278316112387"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="p_1">
<show object="false" label="false" ev="8"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="0.0" y="9.958333333333334" z="-9.958333333333334"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="q">
<show object="false" label="false" ev="8"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="2.4772675074119266" y="-9.909070029647706" z="-14.86360504447156"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Polygon">
<input a0="O" a1="Q" a2="F"/>
<output a0="tOQF" a1="f_2" a2="o_2" a3="q_1"/>
</command>
<element type="polygon" label="tOQF">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false" ev="8"/>
<condition showObject="proof ≟ 2 ∨ proof ≟ 3"/>
<objColor r="255" g="0" b="255" alpha="0.1"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="f_2">
<show object="false" label="false" ev="8"/>
<objColor r="255" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-1.0" y="0.041666666666666664" z="0.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="o_2">
<show object="false" label="false" ev="8"/>
<objColor r="255" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="1.0" y="5.958333333333333" z="-6.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="q_1">
<show object="false" label="false" ev="8"/>
<objColor r="255" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="0.0" y="-6.0" z="0.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Polygon">
<input a0="P" a1="Q" a2="i_T"/>
<output a0="tPQI_T" a1="i_t" a2="p_2" a3="q_2"/>
</command>
<element type="polygon" label="tPQI_T">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false" ev="8"/>
<condition showObject="proof ≟ 2 ∨ proof ≟ 3"/>
<objColor r="0" g="0" b="255" alpha="0.1"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="i_t">
<show object="false" label="false" ev="8"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-2.4772675074119266" y="-0.049263303685626945" z="0.15248278316112387"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="p_2">
<show object="false" label="false" ev="8"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="2.4772675074119226" y="14.760385564996014" z="-14.86360504447151"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="q_2">
<show object="false" label="false" ev="8"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="3.9968028886505635E-15" y="-14.711122261310386" z="-21.7322629141981"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="text6" exp=""od""/>
<element type="text" label="text6">
<show object="true" label="true" ev="40"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.0" size="0" style="1"/>
<startPoint exp="(Q + O_T) / 2"/>
<labelOffset x="-16" y="-43"/>
</element>
<element type="numeric" label="proof">
<value val="3.0"/>
<show object="true" label="false" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="3" absoluteScreenLocation="true" width="200.0" x="160.0" y="299.0" fixed="true" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<expression label="text7" exp=""id""/>
<element type="text" label="text7">
<show object="true" label="true" ev="40"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.0" size="0" style="1"/>
<startPoint exp="(P + i_T) / 2"/>
<labelOffset x="-17" y="1"/>
</element>
<expression label="text8" exp=""c""/>
<element type="text" label="text8">
<show object="true" label="true" ev="40"/>
<objColor r="255" 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="1"/>
<startPoint exp="(O + F) / 2"/>
<labelOffset x="-20" y="-26"/>
</element>
<element type="boolean" label="a">
<value val="false"/>
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="257" y="25"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<caption val="切線法線"/>
</element>
<expression label="text9" exp=""i""/>
<element type="text" label="text9">
<show object="true" label="true" ev="40"/>
<condition showObject="proof > 0"/>
<objColor r="127" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.0" size="0" style="1"/>
<startPoint exp="(O + P) / 2"/>
<labelOffset x="-27" y="-13"/>
</element>
<expression label="text10" exp=""o""/>
<element type="text" label="text10">
<show object="true" label="true" ev="40"/>
<condition showObject="proof > 0"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.0" size="0" style="1"/>
<startPoint exp="(O + Q) / 2"/>
<labelOffset x="-29" y="-15"/>
</element>
<expression label="text11" exp=""公式說明""/>
<element type="text" label="text11">
<show object="true" label="true" ev="43"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<font serif="false" sizeM="1.0" size="0" style="1"/>
<startPoint x="-3.6800000000000006" y="0.25999999999999995" z="1.0"/>
</element>
<expression label="text12" exp=""\frac{c}{od}=\frac{i}{i+o}""/>
<element type="text" label="text12">
<show object="true" label="true" ev="43"/>
<condition showObject="proof > 0"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.0" size="0" style="1"/>
<absoluteScreenLocation x="34" y="324"/>
</element>
<expression label="text13" exp=""\frac{c}{id}=\frac{o}{i+o}""/>
<element type="text" label="text13">
<show object="true" label="true" ev="43"/>
<condition showObject="proof > 1"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.0" size="0" style="1"/>
<startPoint x="-0.31999999999999995" y="-0.3600000000000001" z="1.0"/>
</element>
</construction>
</geogebra>