Section Formula and Barycentric Coordinates in Triangles
2016-11-01 07:07
This GeoGebra applet demonstrates the section formula and barycentric coordinates for a point P relative to a triangle ABC. Using vector and matrix operations, it calculates the linear combination coefficients of P with respect to A, B, and C. The interactive quiz and dynamic verification help learners intuitively understand affine combinations, basis transformations, and the geometric meaning of
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<command name="Segment">
<input a0="Px" a1="P"/>
<output a0="g"/>
</command>
<element type="segment" label="g">
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<condition showObject="showFormula"/>
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<expression label="txtX" exp=""x""/>
<element type="text" label="txtX">
<show object="true" label="false" ev="40"/>
<condition showObject="showFormula"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<bgColor r="255" g="250" b="205" alpha="255"/>
<layer val="1"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="1"/>
<startPoint exp="Px"/>
<labelOffset x="-5" y="6"/>
</element>
<expression label="txtY" exp=""y""/>
<element type="text" label="txtY">
<show object="true" label="false" ev="40"/>
<condition showObject="showFormula"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<bgColor r="255" g="250" b="205" alpha="255"/>
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<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="1"/>
<startPoint exp="Py"/>
<labelOffset x="-27" y="-15"/>
</element>
<expression label="txtProof1" exp=""P=A+\fgcolor{ff0000}{x}\overrightarrow{AB}+\fgcolor{006600}{y} \overrightarrow{AC}""/>
<element type="text" label="txtProof1">
<show object="true" label="false" ev="40"/>
<condition showObject="showFormula"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
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<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="2" size="12" style="1"/>
<absoluteScreenLocation x="7" y="325"/>
</element>
<expression label="txtProof2" exp=""=A+\fgcolor{ff0000}{x}(B-A)+\fgcolor{006600}{y}(C-A)\\ =(1-\fgcolor{ff0000}{x}-\fgcolor{006600}{y})A+(\fgcolor{ff0000}{x})B+(\fgcolor{006600}{y})C""/>
<element type="text" label="txtProof2">
<show object="true" label="false" ev="40"/>
<condition showObject="showFormula"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
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<isLaTeX val="true"/>
<font serif="true" sizeM="2" size="12" style="1"/>
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<element type="boolean" label="showFormula">
<value val="false"/>
<show object="true" label="true"/>
<objColor r="255" g="0" b="0" alpha="0"/>
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<labelMode val="0"/>
<checkbox fixed="true"/>
<ggbscript onUpdate="SetValue[A, If[showFormula, (-2, -2), (0, 0)]] SetValue[B, If[showFormula, ( 1, -1), (3, 1)]] SetValue[C, If[showFormula, (-3, 0), (-1, 2)]] SetValue[P, If[showFormula, ( 3, 1), (2, 3)]]"/>
<caption val="顯示公式"/>
</element>
</construction>
</geogebra>