Exploration of the Fermat Point Problem
2019-09-04 10:42
This GeoGebra applet demonstrates the geometric construction and properties of the Fermat point, which minimizes the sum of distances to a triangle's vertices. Using polygon, rotation, and circle commands, it dynamically shows that when all interior angles are under 120 degrees, the angles between lines connecting the Fermat point to the vertices are exactly 120 degrees. It helps learners intuitiv
In collections
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<expression label="text4" exp=""$\textcolor{blue}{PA}+\textcolor{#0C0}{PB}+\textcolor{#80F}{PC} =$ " + (LaTeX[sPA + sPB + sPC]) + """ />
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<expression label="text5" exp=""将 $\triangle PBC$ 绕 $B$ 点顺转 $60^\circ$ 得 $\triangle BEF$。""/>
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<expression label="text6" exp=""可得 $\textcolor{#00F}{AP}+\textcolor{#0C0}{PB}+\textcolor{#80F}{PC} = \textcolor{#00F}{AP}+\textcolor{#0C0}{PE}+\textcolor{#80F}{EF} \geq \textcolor{red}{AF}$""/>
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<expression label="text9" exp=""此时 $\textcolor{#80F}{\angle PCB} = \textcolor{#F00}{\angle PFB}$，所以 $P, C, F, B$ 共圆。""/>
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<expression label="text12" exp=""此时有 $\textcolor{#00F}{\angle APB} = \textcolor{#0C0}{\angle BPC} = \angle CPA = 120^\circ$""/>
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<expression label="text7" exp=""当 $P, E$ 在 $AF$ 上， $PA+PB+PC$ =$ \sqrt{6}+\sqrt{2}$。 ""/>
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