Arcs and Angles at Centre and Circumference
2015-03-12 02:04
This GeoGebra applet dynamically demonstrates the geometric relationships between arcs, central angles, and inscribed angles in a circle. Through interactive construction, users can observe the ratio of central to inscribed angles subtended by the same arc, and how different arc lengths affect the angles. It utilizes intersections, segments, angle measurements, and dynamic coordinates to help intu
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<coords x="2.3999999999999995" y="0" z="1"/>
<pointSize val="9"/>
<pointStyle val="1"/>
</element>
<command name="Dilate">
<input a0="Intersect[arc2, AngularBisector[D, O, C]]" a1="1.2"/>
<output a0="L2"/>
</command>
<element type="point" label="L2">
<show object="false" label="true" ev="4"/>
<objColor r="160" g="160" b="160" alpha="0"/>
<layer val="8"/>
<labelMode val="0"/>
<coords x="-2.25526228988618" y="-0.8208483439816053" z="1"/>
<pointSize val="9"/>
<pointStyle val="1"/>
</element>
<expression label="Lq" exp="Q + (0.8 * UnitVector[Vector[Q, Intersect[arc2, AngularBisector[D, O, C]]]])" type="point" />
<element type="point" label="Lq">
<show object="false" label="true"/>
<objColor r="160" g="160" b="160" alpha="0"/>
<layer val="8"/>
<labelMode val="0"/>
<coords x="-0.7517540966287267" y="-0.2736161146605351" z="1"/>
<pointSize val="9"/>
<pointStyle val="1"/>
</element>
<expression label="Lp" exp="P + (0.8 * UnitVector[Vector[P, Intersect[arc1, AngularBisector[B, O, A]]]])" type="point" />
<element type="point" label="Lp">
<show object="false" label="true" ev="4"/>
<objColor r="160" g="160" b="160" alpha="0"/>
<layer val="8"/>
<labelMode val="0"/>
<coords x="0.8" y="0" z="1"/>
<pointSize val="9"/>
<pointStyle val="1"/>
</element>
<expression label="textP" exp=""" + (If[Defined[α], Text[α], ""]) + """ />
<element type="text" label="textP">
<show object="true" label="true" ev="8"/>
<condition showObject="showAngle1"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="8"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<startPoint exp="Lp"/>
</element>
<expression label="textQ" exp=""" + (If[Defined[β], Text[β], ""]) + """ />
<element type="text" label="textQ">
<show object="true" label="true" ev="8"/>
<condition showObject="showArc2 ∧ showAngle2"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="8"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<startPoint exp="Lq"/>
</element>
<element type="button" label="button0">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="8"/>
<labelOffset x="869" y="601"/>
<labelMode val="3"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<ggbscript val="SetValue[lang,If[lang==1,2,1]] RunClickScript[buttonLang]"/>
<font serif="false" sizeM="1.4" size="10" style="0"/>
<caption val="中文"/>
</element>
<element type="button" label="buttonLang">
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="8"/>
<labelOffset x="964" y="619"/>
<labelMode val="0"/>
<fixed val="true"/>
<auxiliary val="true"/>
<ggbscript val="SetCaption[button0,If[lang==1,"ENG","中文"]] SetCaption[showAngle1,If[lang==1,"顯示角APB的大小","Show value of angle APB"]] SetCaption[showArc2,If[lang==1,"顯示弧CD","Show arc CD"]] SetCaption[showAngle2,If[lang==1,"顯示角CQD的大小","Show value of angle CQD"]] SetCaption[connectABCD,If[lang==1,"A和C、B和D相接","Connect A and C, B and D"]] SetCaption[showPartition,If[lang==1,"顯示扇形","Show sectors"]] SetCaption[listN,If[lang==1,"將圓周分成 ","Divide circumference into "]] SetCoords[text1, If[lang==1,628,785], 33]"/>
<font serif="false" sizeM="1.4" size="10" style="0"/>
</element>
<element type="numeric" label="lang">
<value val="2"/>
<symbolic val="true" />
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="8"/>
<labelMode val="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<expression label="text1" exp=""" + (If[lang ≟ 1, "等份", "equal portions"]) + """ />
<element type="text" label="text1">
<show object="true" label="true" ev="8"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="2"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<absoluteScreenLocation x="785" y="33"/>
</element>
</construction>
</geogebra>