Geometric Demonstration of the Law of Sines

2015-09-15 04:00

This applet visually demonstrates the Law of Sines. In any triangle, the ratio of a side to the sine of its opposite angle is constant. By constructing a triangle and its circumscribed circle using segments and angles, it illustrates the proportional relationship between side lengths and the sines of opposite angles, helping learners understand the theorem's geometric meaning.

#triangle #trigonometry #geometry-construction #angle #circle

In collections 三角函數

<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.150.0" id="2955e542-f265-4159-903c-5bce97614f82"  xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
	<window width="1336" height="545" />
	<perspectives>
<perspective id="tmp">
	<panes>
		<pane location="" divider="0.1497005988023952" orientation="1" />
	</panes>
	<views>
		<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1,1" size="400" window="643,35,700,486" />
		<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="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="1" size="1126" window="100,100,600,400" />
		<view id="2" visible="true" inframe="false" stylebar="false" location="3" size="200" 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" toolbar="0" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
		<view id="128" visible="false" inframe="true" stylebar="false" location="1" size="480" 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" />
		<view id="43" visible="false" inframe="false" stylebar="false" location="1" size="450" window="50,50,500,500" />
	</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="algebra" />
	<dockBar show="false" east="false" />
</perspective>
	</perspectives>
	<labelingStyle  val="3"/>
	<font  size="12"/>
	<graphicsSettings javaLatexFonts="false"/>
</gui>
<euclidianView>
	<viewNumber viewNo="1"/>
	<size  width="1126" height="543"/>
	<coordSystem xZero="-70.98787499999999" yZero="269.365625" scale="62.08124999999997" yscale="62.08124999999999"/>
	<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" tickDistance="NaN"/>
	<axis id="1" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true" tickDistance="NaN"/>
</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="" author="" date="">
<element type="point" label="A">
	<show object="true" label="true" ev="4"/>
	<objColor r="0" g="0" b="255" alpha="0"/>
	<layer val="0"/>
	<labelOffset x="-8" y="0"/>
	<labelMode val="0"/>
	<animation step="1" speed="1" type="1" playing="false"/>
	<coords x="5.524822309473475" y="0.3924796134098464" z="1"/>
	<pointSize val="3"/>
	<pointStyle val="0"/>
</element>
<element type="point" label="B">
	<show object="true" label="true" ev="4"/>
	<objColor r="0" g="0" b="255" alpha="0"/>
	<layer val="0"/>
	<labelOffset x="-18" y="15"/>
	<labelMode val="0"/>
	<animation step="1" speed="1" type="1" playing="false"/>
	<coords x="2.3" y="-0.22" z="1"/>
	<pointSize val="3"/>
	<pointStyle val="0"/>
</element>
<command name="Segment">
	<input a0="A" a1="B"/>
	<output a0="a"/>
</command>
<element type="segment" label="a">
	<show object="true" label="false" ev="4"/>
	<objColor r="0" g="0" b="0" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<coords x="0.6124796134098464" y="-3.224822309473475" z="-2.1181640189268114"/>
	<lineStyle thickness="2" type="0" typeHidden="1"/>
	<outlyingIntersections val="false"/>
	<keepTypeOnTransform val="true"/>
</element>
<expression label="text1" exp="&quot;sinA=sinD=\frac{\overline{BC}}{\overline{BD}}=\frac{a}{2R}&quot;"/>
<element type="text" label="text1">
	<show object="true" label="false" ev="40"/>
	<objColor r="0" g="0" b="255" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<isLaTeX val="true"/>
	<font serif="true" sizeM="2" size="12" style="0"/>
	<startPoint  x="8.295385080036246" y="3.662387999597303" z="1"/>
</element>
<expression label="text2" exp="&quot;a&quot;"/>
<element type="text" label="text2">
	<show object="true" label="false" ev="40"/>
	<objColor r="255" g="127" b="0" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<isLaTeX val="true"/>
	<font serif="true" sizeM="2" size="12" style="0"/>
	<startPoint exp="E"/>
	<labelOffset x="-15" y="-4"/>
</element>
<expression label="text3" exp="&quot;2R&quot;"/>
<element type="text" label="text3">
	<show object="true" label="false" ev="40"/>
	<objColor r="255" g="127" b="0" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<isLaTeX val="true"/>
	<font serif="true" sizeM="2" size="12" style="0"/>
	<startPoint exp="O"/>
	<labelOffset x="-29" y="-48"/>
</element>
<expression label="text4" exp="&quot;(1)當A為銳角(等弧對等圓周角)&quot;"/>
<element type="text" label="text4">
	<show object="true" label="false" ev="40"/>
	<objColor r="0" g="0" b="255" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<font serif="true" sizeM="2" size="12" style="0"/>
	<startPoint  x="7.763823618242226" y="3.742927615020639" z="1"/>
</element>
<expression label="text1_1" exp="&quot;sinA=sin90°=1=\frac{2R}{2R}=\frac{a}{2R}&quot;"/>
<element type="text" label="text1_1">
	<show object="true" label="false" ev="40"/>
	<objColor r="0" g="0" b="255" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<isLaTeX val="true"/>
	<font serif="true" sizeM="2" size="12" style="0"/>
	<startPoint  x="8.27927715695158" y="1.8905164602839029" z="1"/>
</element>
<expression label="text4_1" exp="&quot;(2)當A為直角(a=2R)&quot;"/>
<element type="text" label="text4_1">
	<show object="true" label="false" ev="40"/>
	<objColor r="0" g="0" b="255" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<isLaTeX val="true"/>
	<font serif="true" sizeM="2" size="12" style="0"/>
	<startPoint  x="7.715499848988225" y="2.3737541528239205" z="1"/>
</element>
<expression label="text1_2" exp="&quot;sinA=sin(180°-∠D)=sinD=\frac{\overline{BC}}{\overline{BD}}=\frac{a}{2R}&quot;"/>
<element type="text" label="text1_2">
	<show object="true" label="false" ev="40"/>
	<objColor r="0" g="0" b="255" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<isLaTeX val="true"/>
	<font serif="true" sizeM="2" size="12" style="0"/>
	<startPoint  x="8.295385080036246" y="0.1508607671398373" z="1"/>
</element>
<expression label="text4_2" exp="&quot;(3)當A為鈍角(∠A+∠D=180°:圓內接四邊形對角互補)&quot;"/>
<element type="text" label="text4_2">
	<show object="true" label="false" ev="40"/>
	<objColor r="0" g="0" b="255" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<isLaTeX val="true"/>
	<font serif="true" sizeM="2" size="12" style="0"/>
	<startPoint  x="7.731607772072891" y="0.6340984596798547" z="1"/>
</element>
<element type="point" label="C">
	<show object="true" label="true" ev="4"/>
	<objColor r="0" g="0" b="255" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<animation step="0.1" speed="1" type="1" playing="false"/>
	<coords x="6.153031309775499" y="-0.2679452330615118" z="1"/>
	<pointSize val="3"/>
	<pointStyle val="0"/>
</element>
<command name="Segment">
	<input a0="B" a1="C"/>
	<output a0="b"/>
</command>
<element type="segment" label="b">
	<show object="true" label="false" ev="4"/>
	<objColor r="0" g="0" b="0" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<coords x="0.0479452330615118" y="3.853031309775499" z="0.7373928521091326"/>
	<lineStyle thickness="2" type="0" typeHidden="1"/>
	<outlyingIntersections val="false"/>
	<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
	<input a0="C" a1="A"/>
	<output a0="c"/>
</command>
<element type="segment" label="c">
	<show object="true" label="false" ev="4"/>
	<objColor r="0" g="0" b="0" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<coords x="-0.6604248464713582" y="-0.6282090003020242" z="3.8952891510946785"/>
	<lineStyle thickness="2" type="0" typeHidden="1"/>
	<outlyingIntersections val="false"/>
	<keepTypeOnTransform val="true"/>
</element>
<command name="Circle">
	<input a0="A" a1="B" a2="C"/>
	<output a0="d"/>
</command>
<element type="conic" label="d">
	<show object="true" 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"/>
	<eigenvectors  x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
	<matrix A0="1" A1="1" A2="14.686392179271158" A3="0" A4="-4.211058069340313" A5="1.4861933188766348"/>
	<eqnStyle style="specific"/>
</element>
<command name="Center">
	<input a0="d"/>
	<output a0="O"/>
</command>
<element type="point" label="O">
	<show object="true" label="true" ev="4"/>
	<objColor r="68" g="68" b="68" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<coords x="4.211058069340313" y="-1.4861933188766348" z="1"/>
	<pointSize val="3"/>
	<pointStyle val="0"/>
</element>
<command name="Line">
	<input a0="B" a1="O"/>
	<output a0="e"/>
</command>
<element type="line" label="e">
	<show object="false" label="false" ev="4"/>
	<objColor r="0" g="0" b="0" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<coords x="1.2661933188766348" y="1.9110580693403127" z="-2.4918118581613906"/>
	<lineStyle thickness="2" type="0" typeHidden="1"/>
	<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
	<input a0="d" a1="e" a2="2"/>
	<output a0="D"/>
</command>
<element type="point" label="D">
	<show object="true" label="true" ev="4"/>
	<objColor r="68" g="68" b="68" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<coords x="6.122116138680626" y="-2.7523866377532706" z="1"/>
	<pointSize val="3"/>
	<pointStyle val="0"/>
</element>
<command name="Segment">
	<input a0="D" a1="C"/>
	<output a0="f"/>
</command>
<element type="segment" label="f">
	<show object="true" label="false" ev="4"/>
	<objColor r="0" g="0" b="255" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<coords x="-2.4844414046917587" y="0.03091517109487274" z="15.295129323095168"/>
	<lineStyle thickness="4" type="15" typeHidden="1"/>
	<outlyingIntersections val="false"/>
	<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
	<input a0="B" a1="D"/>
	<output a0="g"/>
</command>
<element type="segment" label="g">
	<show object="true" label="false" ev="4"/>
	<objColor r="0" g="0" b="255" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<coords x="2.5323866377532704" y="3.8221161386806264" z="-4.983623716322785"/>
	<lineStyle thickness="5" type="0" typeHidden="1"/>
	<outlyingIntersections val="false"/>
	<keepTypeOnTransform val="true"/>
</element>
<command name="Midpoint">
	<input a0="b"/>
	<output a0="E"/>
</command>
<element type="point" label="E">
	<show object="false" label="true" ev="4"/>
	<objColor r="68" g="68" b="68" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<coords x="4.226515654887749" y="-0.24397261653075591" z="1"/>
	<pointSize val="3"/>
	<pointStyle val="0"/>
</element>
<command name="Angle">
	<input a0="B" a1="C" a2="D"/>
	<output a0="β"/>
</command>
<element type="angle" label="β">
	<angleStyle val="1"/>
	<value val="1.5707963267948972"/>
	<show object="true" label="false" ev="4"/>
	<objColor r="255" g="0" b="0" alpha="0.1"/>
	<layer val="0"/>
	<labelMode val="1"/>
	<lineStyle thickness="2" type="0" typeHidden="2"/>
	<arcSize val="30"/>
</element>
<command name="Angle">
	<input a0="C" a1="A" a2="B"/>
	<output a0="γ"/>
</command>
<element type="angle" label="γ">
	<angleStyle val="1"/>
	<value val="4.139676736500042"/>
	<show object="true" label="true" ev="4"/>
	<objColor r="0" g="100" b="0" alpha="0.1"/>
	<layer val="0"/>
	<labelOffset x="-22" y="22"/>
	<labelMode val="1"/>
	<lineStyle thickness="2" type="0" typeHidden="2"/>
	<arcSize val="30"/>
</element>
<command name="Angle">
	<input a0="C" a1="D" a2="B"/>
	<output a0="α"/>
</command>
<element type="angle" label="α">
	<angleStyle val="1"/>
	<value val="0.9980840829102482"/>
	<show object="true" label="true" ev="4"/>
	<objColor r="0" g="100" b="0" alpha="0.1"/>
	<layer val="0"/>
	<labelOffset x="-43" y="17"/>
	<labelMode val="1"/>
	<lineStyle thickness="2" type="0" typeHidden="2"/>
	<arcSize val="30"/>
</element>
<expression label="text5" exp="&quot;∴\frac{a}{sinA}=2R&quot;"/>
<element type="text" label="text5">
	<show object="true" label="false" ev="40"/>
	<objColor r="0" g="0" b="255" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<isLaTeX val="true"/>
	<font serif="true" sizeM="2" size="12" style="0"/>
	<startPoint  x="7.779931541326894" y="-1.1216651565488773" z="1"/>
</element>
<expression label="text5_1" exp="&quot;得正弦定理&quot;"/>
<element type="text" label="text5_1">
	<show object="true" label="false" ev="40"/>
	<objColor r="0" g="0" b="255" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<isLaTeX val="true"/>
	<font serif="true" sizeM="2" size="12" style="0"/>
	<startPoint  x="7.86047115675023" y="-2.329759387898923" z="1"/>
</element>
<expression label="text5_2" exp="&quot;\frac{a}{sinA}=\frac{b}{sinB}=\frac{c}{sinC}=2R&quot;"/>
<element type="text" label="text5_2">
	<show object="true" label="false" ev="40"/>
	<objColor r="0" g="0" b="255" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<isLaTeX val="true"/>
	<font serif="true" sizeM="2" size="12" style="0"/>
	<startPoint  x="9.841745696164304" y="-2.941860465116279" z="1"/>
</element>
</construction>
</geogebra>

Geometric Demonstration of the Law of Sines

Loading the GeoGebra board and drawing, please wait…

<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.150.0" id="2955e542-f265-4159-903c-5bce97614f82"  xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
	<window width="1336" height="545" />
	<perspectives>
<perspective id="tmp">
	<panes>
		<pane location="" divider="0.1497005988023952" orientation="1" />
	</panes>
	<views>
		<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1,1" size="400" window="643,35,700,486" />
		<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="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="1" size="1126" window="100,100,600,400" />
		<view id="2" visible="true" inframe="false" stylebar="false" location="3" size="200" 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" toolbar="0" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
		<view id="128" visible="false" inframe="true" stylebar="false" location="1" size="480" 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" />
		<view id="43" visible="false" inframe="false" stylebar="false" location="1" size="450" window="50,50,500,500" />
	</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="algebra" />
	<dockBar show="false" east="false" />
</perspective>
	</perspectives>
	<labelingStyle  val="3"/>
	<font  size="12"/>
	<graphicsSettings javaLatexFonts="false"/>
</gui>
<euclidianView>
	<viewNumber viewNo="1"/>
	<size  width="1126" height="543"/>
	<coordSystem xZero="-70.98787499999999" yZero="269.365625" scale="62.08124999999997" yscale="62.08124999999999"/>
	<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" tickDistance="NaN"/>
	<axis id="1" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true" tickDistance="NaN"/>
</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="" author="" date="">
<element type="point" label="A">
	<show object="true" label="true" ev="4"/>
	<objColor r="0" g="0" b="255" alpha="0"/>
	<layer val="0"/>
	<labelOffset x="-8" y="0"/>
	<labelMode val="0"/>
	<animation step="1" speed="1" type="1" playing="false"/>
	<coords x="5.524822309473475" y="0.3924796134098464" z="1"/>
	<pointSize val="3"/>
	<pointStyle val="0"/>
</element>
<element type="point" label="B">
	<show object="true" label="true" ev="4"/>
	<objColor r="0" g="0" b="255" alpha="0"/>
	<layer val="0"/>
	<labelOffset x="-18" y="15"/>
	<labelMode val="0"/>
	<animation step="1" speed="1" type="1" playing="false"/>
	<coords x="2.3" y="-0.22" z="1"/>
	<pointSize val="3"/>
	<pointStyle val="0"/>
</element>
<command name="Segment">
	<input a0="A" a1="B"/>
	<output a0="a"/>
</command>
<element type="segment" label="a">
	<show object="true" label="false" ev="4"/>
	<objColor r="0" g="0" b="0" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<coords x="0.6124796134098464" y="-3.224822309473475" z="-2.1181640189268114"/>
	<lineStyle thickness="2" type="0" typeHidden="1"/>
	<outlyingIntersections val="false"/>
	<keepTypeOnTransform val="true"/>
</element>
<expression label="text1" exp="&quot;sinA=sinD=\frac{\overline{BC}}{\overline{BD}}=\frac{a}{2R}&quot;"/>
<element type="text" label="text1">
	<show object="true" label="false" ev="40"/>
	<objColor r="0" g="0" b="255" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<isLaTeX val="true"/>
	<font serif="true" sizeM="2" size="12" style="0"/>
	<startPoint  x="8.295385080036246" y="3.662387999597303" z="1"/>
</element>
<expression label="text2" exp="&quot;a&quot;"/>
<element type="text" label="text2">
	<show object="true" label="false" ev="40"/>
	<objColor r="255" g="127" b="0" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<isLaTeX val="true"/>
	<font serif="true" sizeM="2" size="12" style="0"/>
	<startPoint exp="E"/>
	<labelOffset x="-15" y="-4"/>
</element>
<expression label="text3" exp="&quot;2R&quot;"/>
<element type="text" label="text3">
	<show object="true" label="false" ev="40"/>
	<objColor r="255" g="127" b="0" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<isLaTeX val="true"/>
	<font serif="true" sizeM="2" size="12" style="0"/>
	<startPoint exp="O"/>
	<labelOffset x="-29" y="-48"/>
</element>
<expression label="text4" exp="&quot;(1)當A為銳角(等弧對等圓周角)&quot;"/>
<element type="text" label="text4">
	<show object="true" label="false" ev="40"/>
	<objColor r="0" g="0" b="255" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<font serif="true" sizeM="2" size="12" style="0"/>
	<startPoint  x="7.763823618242226" y="3.742927615020639" z="1"/>
</element>
<expression label="text1_1" exp="&quot;sinA=sin90°=1=\frac{2R}{2R}=\frac{a}{2R}&quot;"/>
<element type="text" label="text1_1">
	<show object="true" label="false" ev="40"/>
	<objColor r="0" g="0" b="255" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<isLaTeX val="true"/>
	<font serif="true" sizeM="2" size="12" style="0"/>
	<startPoint  x="8.27927715695158" y="1.8905164602839029" z="1"/>
</element>
<expression label="text4_1" exp="&quot;(2)當A為直角(a=2R)&quot;"/>
<element type="text" label="text4_1">
	<show object="true" label="false" ev="40"/>
	<objColor r="0" g="0" b="255" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<isLaTeX val="true"/>
	<font serif="true" sizeM="2" size="12" style="0"/>
	<startPoint  x="7.715499848988225" y="2.3737541528239205" z="1"/>
</element>
<expression label="text1_2" exp="&quot;sinA=sin(180°-∠D)=sinD=\frac{\overline{BC}}{\overline{BD}}=\frac{a}{2R}&quot;"/>
<element type="text" label="text1_2">
	<show object="true" label="false" ev="40"/>
	<objColor r="0" g="0" b="255" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<isLaTeX val="true"/>
	<font serif="true" sizeM="2" size="12" style="0"/>
	<startPoint  x="8.295385080036246" y="0.1508607671398373" z="1"/>
</element>
<expression label="text4_2" exp="&quot;(3)當A為鈍角(∠A+∠D=180°:圓內接四邊形對角互補)&quot;"/>
<element type="text" label="text4_2">
	<show object="true" label="false" ev="40"/>
	<objColor r="0" g="0" b="255" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<isLaTeX val="true"/>
	<font serif="true" sizeM="2" size="12" style="0"/>
	<startPoint  x="7.731607772072891" y="0.6340984596798547" z="1"/>
</element>
<element type="point" label="C">
	<show object="true" label="true" ev="4"/>
	<objColor r="0" g="0" b="255" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<animation step="0.1" speed="1" type="1" playing="false"/>
	<coords x="6.153031309775499" y="-0.2679452330615118" z="1"/>
	<pointSize val="3"/>
	<pointStyle val="0"/>
</element>
<command name="Segment">
	<input a0="B" a1="C"/>
	<output a0="b"/>
</command>
<element type="segment" label="b">
	<show object="true" label="false" ev="4"/>
	<objColor r="0" g="0" b="0" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<coords x="0.0479452330615118" y="3.853031309775499" z="0.7373928521091326"/>
	<lineStyle thickness="2" type="0" typeHidden="1"/>
	<outlyingIntersections val="false"/>
	<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
	<input a0="C" a1="A"/>
	<output a0="c"/>
</command>
<element type="segment" label="c">
	<show object="true" label="false" ev="4"/>
	<objColor r="0" g="0" b="0" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<coords x="-0.6604248464713582" y="-0.6282090003020242" z="3.8952891510946785"/>
	<lineStyle thickness="2" type="0" typeHidden="1"/>
	<outlyingIntersections val="false"/>
	<keepTypeOnTransform val="true"/>
</element>
<command name="Circle">
	<input a0="A" a1="B" a2="C"/>
	<output a0="d"/>
</command>
<element type="conic" label="d">
	<show object="true" 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"/>
	<eigenvectors  x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
	<matrix A0="1" A1="1" A2="14.686392179271158" A3="0" A4="-4.211058069340313" A5="1.4861933188766348"/>
	<eqnStyle style="specific"/>
</element>
<command name="Center">
	<input a0="d"/>
	<output a0="O"/>
</command>
<element type="point" label="O">
	<show object="true" label="true" ev="4"/>
	<objColor r="68" g="68" b="68" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<coords x="4.211058069340313" y="-1.4861933188766348" z="1"/>
	<pointSize val="3"/>
	<pointStyle val="0"/>
</element>
<command name="Line">
	<input a0="B" a1="O"/>
	<output a0="e"/>
</command>
<element type="line" label="e">
	<show object="false" label="false" ev="4"/>
	<objColor r="0" g="0" b="0" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<coords x="1.2661933188766348" y="1.9110580693403127" z="-2.4918118581613906"/>
	<lineStyle thickness="2" type="0" typeHidden="1"/>
	<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
	<input a0="d" a1="e" a2="2"/>
	<output a0="D"/>
</command>
<element type="point" label="D">
	<show object="true" label="true" ev="4"/>
	<objColor r="68" g="68" b="68" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<coords x="6.122116138680626" y="-2.7523866377532706" z="1"/>
	<pointSize val="3"/>
	<pointStyle val="0"/>
</element>
<command name="Segment">
	<input a0="D" a1="C"/>
	<output a0="f"/>
</command>
<element type="segment" label="f">
	<show object="true" label="false" ev="4"/>
	<objColor r="0" g="0" b="255" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<coords x="-2.4844414046917587" y="0.03091517109487274" z="15.295129323095168"/>
	<lineStyle thickness="4" type="15" typeHidden="1"/>
	<outlyingIntersections val="false"/>
	<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
	<input a0="B" a1="D"/>
	<output a0="g"/>
</command>
<element type="segment" label="g">
	<show object="true" label="false" ev="4"/>
	<objColor r="0" g="0" b="255" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<coords x="2.5323866377532704" y="3.8221161386806264" z="-4.983623716322785"/>
	<lineStyle thickness="5" type="0" typeHidden="1"/>
	<outlyingIntersections val="false"/>
	<keepTypeOnTransform val="true"/>
</element>
<command name="Midpoint">
	<input a0="b"/>
	<output a0="E"/>
</command>
<element type="point" label="E">
	<show object="false" label="true" ev="4"/>
	<objColor r="68" g="68" b="68" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<coords x="4.226515654887749" y="-0.24397261653075591" z="1"/>
	<pointSize val="3"/>
	<pointStyle val="0"/>
</element>
<command name="Angle">
	<input a0="B" a1="C" a2="D"/>
	<output a0="β"/>
</command>
<element type="angle" label="β">
	<angleStyle val="1"/>
	<value val="1.5707963267948972"/>
	<show object="true" label="false" ev="4"/>
	<objColor r="255" g="0" b="0" alpha="0.1"/>
	<layer val="0"/>
	<labelMode val="1"/>
	<lineStyle thickness="2" type="0" typeHidden="2"/>
	<arcSize val="30"/>
</element>
<command name="Angle">
	<input a0="C" a1="A" a2="B"/>
	<output a0="γ"/>
</command>
<element type="angle" label="γ">
	<angleStyle val="1"/>
	<value val="4.139676736500042"/>
	<show object="true" label="true" ev="4"/>
	<objColor r="0" g="100" b="0" alpha="0.1"/>
	<layer val="0"/>
	<labelOffset x="-22" y="22"/>
	<labelMode val="1"/>
	<lineStyle thickness="2" type="0" typeHidden="2"/>
	<arcSize val="30"/>
</element>
<command name="Angle">
	<input a0="C" a1="D" a2="B"/>
	<output a0="α"/>
</command>
<element type="angle" label="α">
	<angleStyle val="1"/>
	<value val="0.9980840829102482"/>
	<show object="true" label="true" ev="4"/>
	<objColor r="0" g="100" b="0" alpha="0.1"/>
	<layer val="0"/>
	<labelOffset x="-43" y="17"/>
	<labelMode val="1"/>
	<lineStyle thickness="2" type="0" typeHidden="2"/>
	<arcSize val="30"/>
</element>
<expression label="text5" exp="&quot;∴\frac{a}{sinA}=2R&quot;"/>
<element type="text" label="text5">
	<show object="true" label="false" ev="40"/>
	<objColor r="0" g="0" b="255" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<isLaTeX val="true"/>
	<font serif="true" sizeM="2" size="12" style="0"/>
	<startPoint  x="7.779931541326894" y="-1.1216651565488773" z="1"/>
</element>
<expression label="text5_1" exp="&quot;得正弦定理&quot;"/>
<element type="text" label="text5_1">
	<show object="true" label="false" ev="40"/>
	<objColor r="0" g="0" b="255" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<isLaTeX val="true"/>
	<font serif="true" sizeM="2" size="12" style="0"/>
	<startPoint  x="7.86047115675023" y="-2.329759387898923" z="1"/>
</element>
<expression label="text5_2" exp="&quot;\frac{a}{sinA}=\frac{b}{sinB}=\frac{c}{sinC}=2R&quot;"/>
<element type="text" label="text5_2">
	<show object="true" label="false" ev="40"/>
	<objColor r="0" g="0" b="255" alpha="0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<isLaTeX val="true"/>
	<font serif="true" sizeM="2" size="12" style="0"/>
	<startPoint  x="9.841745696164304" y="-2.941860465116279" z="1"/>
</element>
</construction>
</geogebra>