Comprehensive Construction of Circles, Lines, and Symmetry Transformations
2019-05-12 02:12
This applet demonstrates the comprehensive construction of lines, circles, angle bisectors, and symmetry transformations in plane geometry. Using GeoGebra tools, it illustrates how to utilize geometric operations such as lines, orthogonal lines, perpendicular bisectors, and mirror reflections to explore special points generated by the intersection of circles and lines. This construction covers fun
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<expression label="text1" exp=""题：在△ABC中，作∠BAC平分线交BC于点D，作AH⊥BC于H点，AD中垂线分别交ΔABH外接圆和ΔACH外接圆于点E、F，求证：D、E、F、H四点共圆""/>
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<expression label="text2" exp=""解：设EF交AD于点G，交AH于点J，以A为反演中心，\sqrt{AD·AG}为反演半径作原图像的反演，记任一点X(X∈{A,B,C,D,E,F,G,H})经反演后至X'""/>
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<expression label="text3" exp=""则由于∠AGJ=∠AHD=90°，AG·AD=AJ·AH，从而反演后D'与G、G'与D、H'与J、J与H'重合""/>
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<expression label="text4" exp=""而且由反演性质，直线BC→⊙AD'H'，直线EF→⊙AG'J'，其中由于∠AJ'G'=90°，D'为AG'中点，则⊙AG'J'的圆心为点D'，即⊙D'过点E'、F'""/>
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<expression label="text5" exp=""且⊙ABH→直线B'H'，⊙ACH→直线C'H',由于∠AB'E'=∠AEB=180°-∠AHB=90°，则AB'⊥B'H'
同理AC'⊥C'H'""/>
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<expression label="text6" exp=""连接D'E'、D'F'、B'D'、C'D'，由于AD平分∠BAC，∠B'AD'=∠BAD=∠CAD=∠C'AD'，而A、B'、C'、D'四点共圆，则B'D'=C'D'""/>
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<expression label="text7" exp=""又D'E'=D'F'为⊙D'半径，则D、E、F、H四点共圆⇔D'、E'、F'、H'四点共圆⇔∠D'E'H'=∠D'F'H'⇔∠B'E'D'=∠C'F'D'⇔ΔB'E'D'≅ΔC'F'D'""/>
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<expression label="text8" exp=""而∠D'B'E'=∠BB'E'+∠BB'G'=90°+∠AC'D'=∠AC'D'+∠AC'F'=∠D'C'F'且由于此角大小为90°+∠AC'D'>90°为钝角""/>
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<expression label="text9" exp=""又B'D'=C'F'，D'E'=D'F'，从而满足SSA型全等判定，从而ΔB'E'D'≅ΔC'F'D'⇔D、E、F、H四点共圆，得证""/>
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