三维球面几何与向量正交投影演示
2020-12-05 14:42
本GeoGebra范例演示了三维空间中的球面几何与向量投影关系。通过构建球体、赤道圆及空间平面，展示了空间点在球面上的位置及其向特定平面或轴的正交投影。范例利用线段和角计算并可视化了空间向量及其投影向量之间的夹角和长度关系，结合勾股定理深入解析三维向量的坐标分解与空间几何性质，适用于立体几何与三维向量教学。
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.688.0" app="classic" platform="w" id="3ce7f76e-7072-4743-9fd5-5fff4fe81bc6" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="900" height="700" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.30333333333333334" orientation="1" />
</panes>
<views>
<view id="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 || 1008 1009 || 66 68 || 6" visible="false" inframe="false" stylebar="false" location="1,3,3" size="300" 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="true" location="1,1" size="300" window="100,100,600,400" />
<view id="4097" visible="false" inframe="false" stylebar="true" location="1,1" size="337" window="806,334,700,550" />
<view id="1" visible="true" inframe="false" stylebar="false" location="3" size="273" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="true" location="3" size="197" tab="ALGEBRA" window="100,100,250,400" />
<view id="16" visible="false" inframe="false" stylebar="false" location="1" size="336" 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="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" />
<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="true" inframe="false" stylebar="true" location="1" size="619" window="100,100,600,400" />
</views>
<toolbar show="false" items="0 39 | 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 68 | 25 52 60 61 | 40 41 42 , 27 28 35 , 6" position="1" help="false" />
<input show="false" cmd="true" top="false" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="12"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="273" height="700"/>
<coordSystem xZero="137.44661157024782" yZero="172.74449896694247" scale="53.87215909090906" yscale="53.87215909090928"/>
<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="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>
<uses3D val="true"/>
<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"/>
<euclidianView3D>
<coordSystem xZero="0.2275107180293099" yZero="0.008517649369290337" zZero="-0.16548618773864351" scale="239.71814125666592" xAngle="20" zAngle="-688.7291766187086"/>
<evSettings axes="true" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" gridType="0"/>
<axis id="0" show="true" label="" unitLabel="" tickStyle="2" showNumbers="false"/>
<axis id="1" show="true" label="" unitLabel="" tickStyle="2" showNumbers="false"/>
<axis id="2" show="true" label="" unitLabel="" tickStyle="2" showNumbers="false"/>
<plate show="false"/>
<bgColor r="255" g="255" b="255"/>
<clipping use="false" show="false" size="2"/>
<projection type="1"/>
</euclidianView3D>
<construction title="" author="" date="">
<element type="point3d" label="A">
<show object="true" label="false" ev="4"/>
<objColor r="64" g="64" b="64" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
<coords x="0" y="0" z="0" w="1"/>
<pointSize val="5"/>
</element>
<command name="Sphere">
<input a0="A" a1="1"/>
<output a0="s"/>
</command>
<element type="quadric" label="s">
<show object="true" label="false" ev="4"/>
<objColor r="102" g="204" b="255" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eigenvectors x0="1" y0="0" z0="0" x1="0" y1="1" z1="0" x2="0" y2="0" z2="1"/>
<matrix A0="1" A1="1" A2="1" A3="-1" A4="0" A5="0" A6="0" A7="0" A8="0" A9="0"/>
<eqnStyle style="specific"/>
</element>
<expression label="I" exp="(1, 0, 0)" type="point"/>
<element type="point3d" label="I">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-53" y="-23"/>
<labelMode val="3"/>
<fixed val="true"/>
<coords x="1" y="0" z="0" w="1"/>
<pointSize val="1"/>
</element>
<command name="Circle">
<input a0="A" a1="I" a2="zAxis"/>
<output a0="equatorial"/>
</command>
<element type="conic3d" label="equatorial">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="-1" A3="0" A4="0" A5="0"/>
<eqnStyle style="parametric"/>
</element>
<element type="point3d" label="K">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="6" y="-30"/>
<labelMode val="3"/>
<fixed val="true"/>
<coords x="0" y="0" z="1" w="1"/>
<pointSize val="1"/>
</element>
<element type="point3d" label="J">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="15" y="-8"/>
<labelMode val="3"/>
<fixed val="true"/>
<coords x="0" y="1" z="0" w="1"/>
<pointSize val="1"/>
<caption val="(0,1,0)"/>
</element>
<expression label="P" exp="((cos(24°) * cos(121°)), (cos(24°) * sin(121°)), sin(24°))" type="point"/>
<element type="point3d" label="P">
<show object="true" label="true" ev="4"/>
<condition showObject="n ≟ 1"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<selectionAllowed val="false"/>
<coords x="-0.47051069384706967" y="0.7830612939961841" z="0.4067366430758002" w="1"/>
<pointSize val="5"/>
</element>
<command name="Segment">
<input a0="A" a1="P"/>
<output a0="sOP"/>
</command>
<element type="segment3d" label="sOP">
<show object="true" label="false" ev="4"/>
<condition showObject="n ≟ 1"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="text1" exp=""x軸""/>
<element type="text" label="text1">
<show object="true" label="false" ev="37"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="0"/>
<startPoint exp="I"/>
<labelOffset x="-75" y="28"/>
</element>
<expression label="text2" exp=""y軸	""/>
<element type="text" label="text2">
<show object="true" label="false" ev="37"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="0"/>
<startPoint exp="J"/>
<labelOffset x="77" y="22"/>
</element>
<expression label="text3" exp=""z軸""/>
<element type="text" label="text3">
<show object="true" label="false" ev="37"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="0"/>
<startPoint exp="K"/>
<labelOffset x="18" y="-49"/>
</element>
<command name="Plane">
<input a0="A" a1="I" a2="J"/>
<output a0="E_{1}"/>
</command>
<element type="plane3d" label="E_{1}">
<show object="false" label="false" ev="4"/>
<objColor r="99" g="219" b="219" alpha="0.5"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0" y="0" z="1" w="0"/>
<fading val="0.1"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
</element>
<command name="OrthogonalLine">
<input a0="P" a1="E_{1}"/>
<output a0="L_{PE}"/>
</command>
<element type="line3d" label="L_{PE}">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<coords ox="-0.47051069384706967" oy="0.7830612939961841" oz="0.4067366430758002" ow="1" vx="0" vy="0" vz="1" vw="0"/>
<eqnStyle style="parametric" parameter="λ"/>
</element>
<command name="Intersect">
<input a0="L_{PE}" a1="E_{1}"/>
<output a0="P'"/>
</command>
<element type="point3d" label="P'">
<show object="true" label="true" ev="4"/>
<condition showObject="n ≟ 1"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.47051069384706967" y="0.7830612939961841" z="0" w="1"/>
<pointSize val="3"/>
</element>
<command name="Segment">
<input a0="P" a1="P'"/>
<output a0="sPP'"/>
</command>
<element type="segment3d" label="sPP'">
<show object="true" label="false" ev="4"/>
<condition showObject="n ≟ 1"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="A" a1="P'"/>
<output a0="sOP'"/>
</command>
<element type="segment3d" label="sOP'">
<show object="true" label="false" ev="4"/>
<condition showObject="n ≟ 1"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="OrthogonalLine">
<input a0="P'" a1="xAxis" a2="space"/>
<output a0="L_{P'x}"/>
</command>
<element type="line3d" label="L_{P'x}">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<coords ox="-0.47051069384706967" oy="0.7830612939961841" oz="0" ow="1" vx="0" vy="1" vz="0" vw="0"/>
<eqnStyle style="parametric" parameter="λ"/>
</element>
<command name="Intersect">
<input a0="L_{P'x}" a1="xAxis"/>
<output a0="P''"/>
</command>
<element type="point3d" label="P''">
<show object="true" label="true" ev="4"/>
<condition showObject="n ≟ 1"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.47051069384706967" y="0" z="0" w="1"/>
<pointSize val="3"/>
</element>
<command name="Segment">
<input a0="P'" a1="P''"/>
<output a0="sP'P''"/>
</command>
<element type="segment3d" label="sP'P''">
<show object="true" label="false" ev="4"/>
<condition showObject="n ≟ 1"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Angle">
<input a0="P" a1="A" a2="P'"/>
<output a0="φ_{1}"/>
</command>
<element type="angle3d" label="φ_{1}">
<angleStyle val="1"/>
<value val="0.4188790204786389"/>
<show object="true" label="true" ev="4"/>
<condition showObject="n ≟ 1"/>
<objColor r="255" g="153" b="51" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelOffset x="22" y="-8"/>
<labelMode val="1"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="50"/>
</element>
<element type="numeric" label="n">
<value val="1"/>
<show object="true" label="true"/>
<objColor r="0" g="153" b="255" alpha="0.10000000149011612"/>
<bgColor r="0" g="153" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="1" max="2" absoluteScreenLocation="true" width="100" x="91" y="86" 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="text4" exp=""1.\ 臺灣地理中心碑\\2.\ 51區""/>
<element type="text" label="text4">
<show object="true" label="false" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="2" size="12" style="0"/>
<startPoint x="-2.139748984865264" y="-1.300502030269472" z="1"/>
</element>
<expression label="Q" exp="((cos(37°) * cos(-116°)), (cos(37°) * sin(-116°)), sin(37°))" type="point"/>
<element type="point3d" label="Q">
<show object="true" label="true" ev="4"/>
<condition showObject="n ≟ 2"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<selectionAllowed val="false"/>
<coords x="-0.3500987644059116" y="-0.7178088415936053" z="0.6018150231520483" w="1"/>
<pointSize val="5"/>
</element>
<command name="Segment">
<input a0="A" a1="Q"/>
<output a0="sOQ"/>
</command>
<element type="segment3d" label="sOQ">
<show object="true" label="false" ev="4"/>
<condition showObject="n ≟ 2"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="OrthogonalLine">
<input a0="Q" a1="E_{1}"/>
<output a0="L_{QE}"/>
</command>
<element type="line3d" label="L_{QE}">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<coords ox="-0.3500987644059116" oy="-0.7178088415936053" oz="0.6018150231520483" ow="1" vx="0" vy="0" vz="1" vw="0"/>
<eqnStyle style="parametric" parameter="λ"/>
</element>
<command name="Intersect">
<input a0="L_{QE}" a1="E_{1}"/>
<output a0="Q'"/>
</command>
<element type="point3d" label="Q'">
<show object="true" label="true" ev="4"/>
<condition showObject="n ≟ 2"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.3500987644059116" y="-0.7178088415936053" z="0" w="1"/>
<pointSize val="3"/>
</element>
<command name="Segment">
<input a0="A" a1="Q'"/>
<output a0="sOQ'"/>
</command>
<element type="segment3d" label="sOQ'">
<show object="true" label="false" ev="4"/>
<condition showObject="n ≟ 2"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="OrthogonalLine">
<input a0="Q'" a1="xAxis" a2="space"/>
<output a0="L_{Q'x}"/>
</command>
<element type="line3d" label="L_{Q'x}">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<coords ox="-0.3500987644059116" oy="-0.7178088415936053" oz="0" ow="1" vx="0" vy="-1" vz="0" vw="0"/>
<eqnStyle style="parametric" parameter="λ"/>
</element>
<command name="Intersect">
<input a0="xAxis" a1="L_{Q'x}"/>
<output a0="Q''"/>
</command>
<element type="point3d" label="Q''">
<show object="true" label="true" ev="4"/>
<condition showObject="n ≟ 2"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelOffset x="-2" y="-8"/>
<labelMode val="0"/>
<coords x="-0.3500987644059116" y="0" z="0" w="1"/>
<pointSize val="3"/>
</element>
<command name="Angle">
<input a0="Q" a1="A" a2="Q'"/>
<output a0="φ_{2}"/>
</command>
<element type="angle3d" label="φ_{2}">
<angleStyle val="1"/>
<value val="0.6457718232379018"/>
<show object="true" label="true" ev="4"/>
<condition showObject="n ≟ 2"/>
<objColor r="255" g="153" b="51" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelOffset x="-44" y="-17"/>
<labelMode val="1"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="50"/>
</element>
<command name="Segment">
<input a0="Q" a1="Q'"/>
<output a0="sQQ'"/>
</command>
<element type="segment3d" label="sQQ'">
<show object="true" label="false" ev="4"/>
<condition showObject="n ≟ 2"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="Q'" a1="Q''"/>
<output a0="sO'O''"/>
</command>
<element type="segment3d" label="sO'O''">
<show object="true" label="false" ev="4"/>
<condition showObject="n ≟ 2"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Angle">
<input a0="I" a1="A" a2="Q'"/>
<output a0="λ_{2}"/>
</command>
<element type="angle3d" label="λ_{2}">
<angleStyle val="1"/>
<value val="2.0245819323134224"/>
<show object="true" label="true" ev="4"/>
<condition showObject="n ≟ 2"/>
<objColor r="0" g="153" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelOffset x="-37" y="8"/>
<labelMode val="1"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="50"/>
</element>
<command name="Angle">
<input a0="I" a1="A" a2="P'"/>
<output a0="λ_{1}"/>
</command>
<element type="angle3d" label="λ_{1}">
<angleStyle val="1"/>
<value val="2.111848394913139"/>
<show object="true" label="true" ev="4"/>
<condition showObject="n ≟ 1"/>
<objColor r="0" g="153" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelOffset x="-4" y="10"/>
<labelMode val="1"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="50"/>
</element>
<element type="point" label="B">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="-1.981437536254812" y="-8.958128988029337" z="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<element type="point" label="C">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="2.018562463745188" y="-8.958128988029337" z="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<element type="image" label="pic1">
<file name="e2e659a50f30c34f02085e1811d27068/1.jpg"/>
<inBackground val="false"/>
<startPoint number="0" exp="B"/>
<startPoint number="1" exp="C"/>
<show object="true" label="false"/>
<condition showObject="n ≟ 1"/>
<objColor r="0" g="0" b="0" alpha="1"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<expression label="text5" exp=""\sin24°""/>
<element type="text" label="text5">
<show object="true" label="false" ev="37"/>
<condition showObject="n ≟ 1"/>
<objColor r="255" g="102" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="0"/>
<startPoint exp="mPP'"/>
<labelOffset x="7" y="-12"/>
</element>
<expression label="text6" exp=""\cos24°""/>
<element type="text" label="text6">
<show object="true" label="false" ev="37"/>
<condition showObject="n ≟ 1"/>
<objColor r="255" g="102" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="0"/>
<startPoint exp="mAP'"/>
<labelOffset x="-9" y="3"/>
</element>
<expression label="text7" exp=""\cos24°\cos121°""/>
<element type="text" label="text7">
<show object="true" label="false" ev="37"/>
<condition showObject="n ≟ 1"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="0"/>
<startPoint exp="mAP''"/>
<labelOffset x="-76" y="-24"/>
</element>
<expression label="text8" exp=""\cos24°\sin121°""/>
<element type="text" label="text8">
<show object="true" label="false" ev="37"/>
<condition showObject="n ≟ 1"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="0"/>
<startPoint exp="mP'P''"/>
<labelOffset x="-52" y="-32"/>
</element>
<command name="Midpoint">
<input a0="P" a1="P'"/>
<output a0="mPP'"/>
</command>
<element type="point3d" label="mPP'">
<show object="false" label="false" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.47051069384706967" y="0.7830612939961841" z="0.2033683215379001" w="1"/>
<pointSize val="3"/>
</element>
<command name="Midpoint">
<input a0="P'" a1="P''"/>
<output a0="mP'P''"/>
</command>
<element type="point3d" label="mP'P''">
<show object="false" label="false" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.47051069384706967" y="0.39153064699809204" z="0" w="1"/>
<pointSize val="3"/>
</element>
<command name="Midpoint">
<input a0="A" a1="P''"/>
<output a0="mAP''"/>
</command>
<element type="point3d" label="mAP''">
<show object="false" label="false" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelOffset x="-31" y="-13"/>
<labelMode val="0"/>
<coords x="-0.23525534692353484" y="0" z="0" w="1"/>
<pointSize val="3"/>
</element>
<command name="Midpoint">
<input a0="A" a1="P'"/>
<output a0="mAP'"/>
</command>
<element type="point3d" label="mAP'">
<show object="false" label="false" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.23525534692353484" y="0.39153064699809204" z="0" w="1"/>
<pointSize val="3"/>
</element>
<element type="point" label="D">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="-1.981437536254812" y="-7.883879133048577" z="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<element type="point" label="E">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="2.018562463745186" y="-7.883879133048577" z="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<element type="image" label="pic2">
<file name="09501de468fc6d3d39b7129fbfe30973/2.jpg"/>
<inBackground val="false"/>
<startPoint number="0" exp="D"/>
<startPoint number="1" exp="E"/>
<show object="true" label="false"/>
<condition showObject="n ≟ 2"/>
<objColor r="0" g="0" b="0" alpha="1"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<expression label="text9" exp=""\cos37°""/>
<element type="text" label="text9">
<show object="true" label="false" ev="37"/>
<condition showObject="n ≟ 2"/>
<objColor r="255" g="102" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="0"/>
<startPoint exp="mAQ'"/>
<labelOffset x="-71" y="-8"/>
</element>
<expression label="text10" exp=""\sin37°""/>
<element type="text" label="text10">
<show object="true" label="false" ev="37"/>
<condition showObject="n ≟ 2"/>
<objColor r="255" g="102" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="0"/>
<startPoint exp="mQQ'"/>
<labelOffset x="-59" y="-3"/>
</element>
<expression label="text11" exp=""\cos37°\cos(-116°)""/>
<element type="text" label="text11">
<show object="true" label="false" ev="37"/>
<condition showObject="n ≟ 2"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="0"/>
<startPoint exp="mAQ''"/>
<labelOffset x="1" y="-12"/>
</element>
<expression label="text12" exp=""\cos37°\sin(-116°)""/>
<element type="text" label="text12">
<show object="true" label="false" ev="37"/>
<condition showObject="n ≟ 2"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="0"/>
<startPoint exp="mQ'Q''"/>
<labelOffset x="-73" y="-33"/>
</element>
<command name="Midpoint">
<input a0="Q" a1="Q'"/>
<output a0="mQQ'"/>
</command>
<element type="point3d" label="mQQ'">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.3500987644059116" y="-0.7178088415936053" z="0.30090751157602413" w="1"/>
<pointSize val="3"/>
</element>
<command name="Midpoint">
<input a0="Q'" a1="Q''"/>
<output a0="mQ'Q''"/>
</command>
<element type="point3d" label="mQ'Q''">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.3500987644059116" y="-0.3589044207968026" z="0" w="1"/>
<pointSize val="3"/>
</element>
<command name="Midpoint">
<input a0="A" a1="Q''"/>
<output a0="mAQ''"/>
</command>
<element type="point3d" label="mAQ''">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.1750493822029558" y="0" z="0" w="1"/>
<pointSize val="3"/>
</element>
<command name="Midpoint">
<input a0="A" a1="Q'"/>
<output a0="mAQ'"/>
</command>
<element type="point3d" label="mAQ'">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.1750493822029558" y="-0.3589044207968026" z="0" w="1"/>
<pointSize val="3"/>
</element>
</construction>
</geogebra>