Derousseau's Generalization of the Malfatti circles

Test Case in ETC

The test case used in “Encyclopedis of Triangle Centers” to search for triangle centers.

\(a:b:c=6:9:13\).

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Malfatti circles

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Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.214285714286&{}:{}&0.321428571429&{}:{}&0.464285714286&, \\ P_{\mathbf{5}}&{}\approx{}&0.995638119502&{}:{}&0.000183761145&{}:{}&0.004178119353&, \\ P^-_{\mathbf{5}}&{}\approx{}&1.038934456102&{}:{}&-0.017617071545&{}:{}&-0.021317384557&, \\ P^+_{\mathbf{5}}&{}\approx{}&0.956661348297&{}:{}&0.016208650026&{}:{}&0.027130001676&, \\ Q_{\mathbf{5}}&{}\approx{}&0.857603143946&{}:{}&-0.014737453291&{}:{}&0.157134309346&, \\ I^\prime_{\mathbf{5}}&{}\approx{}&0.899286975663&{}:{}&0.014963046273&{}:{}&0.085749978064&, \end{alignedat} \]
\(I\) Incenter
\(P_{\mathbf{5}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{5}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{5}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{5}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{5}}\) Radical center of the Malfatti circles
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Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5}}&{}\approx{}&0.959779082061&{}:{}&0.016454011884&{}:{}&0.023766906055&,\\B^\prime_{\mathbf{5}}&{}\approx{}&0.594177018390&{}:{}&-0.881560558236&{}:{}&1.287383539846&,\\C^\prime_{\mathbf{5}}&{}\approx{}&1.419039168828&{}:{}&2.128558753243&{}:{}&-2.547597922071&, \\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.837625355554&{}:{}&0.006840662084&{}:{}&0.155533982361&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.963301984604&{}:{}&0.032655592164&{}:{}&0.004042423232&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.953151902146&{}:{}&0.000175919625&{}:{}&0.046672178229&, \\ A^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&-0.438795806075&{}:{}&0.651026653308&{}:{}&0.787769152767&,\\B^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&1.003386395763&{}:{}&0.017201595647&{}:{}&-0.020587991410&,\\C^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&0.992254434996&{}:{}&-0.016825524719&{}:{}&0.024571089722&, \\ A^*_{\mathbf{5}}&{}\approx{}&0.000000000000&{}:{}&-0.103495636770&{}:{}&1.103495636770&,\\B^*_{\mathbf{5}}&{}\approx{}&0.845147817462&{}:{}&0.000000000000&{}:{}&0.154852182538&,\\C^*_{\mathbf{5}}&{}\approx{}&1.017484936752&{}:{}&-0.017484936752&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5}}}{B^\prime_{\mathbf{5}}}{C^\prime_{\mathbf{5}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5}}}{B^{\prime\prime}_{\mathbf{5}}}{C^{\prime\prime}_{\mathbf{5}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5}}}{B^{\prime\prime\prime}_{\mathbf{5}}}{C^{\prime\prime\prime}_{\mathbf{5}}}\)
\(\triangle{A^*_{\mathbf{5}}}{B^*_{\mathbf{5}}}{C^*_{\mathbf{5}}}\)
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Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5}}}}&{}\approx{}&0.051190259195&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5}}}}&{}\approx{}&2.772826085822&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5}}}}&{}\approx{}&6.622182787866&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.214285714286&{}:{}&0.321428571429&{}:{}&0.464285714286&,\\ A^\prime_{\mathbf{5}}&{}\approx{}&0.959779082061&{}:{}&0.016454011884&{}:{}&0.023766906055&,\\B^\prime_{\mathbf{5}}&{}\approx{}&0.594177018390&{}:{}&-0.881560558236&{}:{}&1.287383539846&,\\C^\prime_{\mathbf{5}}&{}\approx{}&1.419039168828&{}:{}&2.128558753243&{}:{}&-2.547597922071&. \end{alignedat} \]
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First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5}}&{}\approx{}&0.995638119502&{}:{}&0.000183761145&{}:{}&0.004178119353&,\\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.837625355554&{}:{}&0.006840662084&{}:{}&0.155533982361&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.963301984604&{}:{}&0.032655592164&{}:{}&0.004042423232&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.953151902146&{}:{}&0.000175919625&{}:{}&0.046672178229&. \end{alignedat} \]
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First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5}}&{}\approx{}&1.038934456102&{}:{}&-0.017617071545&{}:{}&-0.021317384557&,\\ A^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&-0.438795806075&{}:{}&0.651026653308&{}:{}&0.787769152767&,\\B^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&1.003386395763&{}:{}&0.017201595647&{}:{}&-0.020587991410&,\\C^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&0.992254434996&{}:{}&-0.016825524719&{}:{}&0.024571089722&. \end{alignedat} \]
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Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5}}&{}\approx{}&0.956661348297&{}:{}&0.016208650026&{}:{}&0.027130001676&,\\ A^\prime_{\mathbf{5}}&{}\approx{}&0.959779082061&{}:{}&0.016454011884&{}:{}&0.023766906055&,\\B^\prime_{\mathbf{5}}&{}\approx{}&0.594177018390&{}:{}&-0.881560558236&{}:{}&1.287383539846&,\\C^\prime_{\mathbf{5}}&{}\approx{}&1.419039168828&{}:{}&2.128558753243&{}:{}&-2.547597922071&,\\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.837625355554&{}:{}&0.006840662084&{}:{}&0.155533982361&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.963301984604&{}:{}&0.032655592164&{}:{}&0.004042423232&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.953151902146&{}:{}&0.000175919625&{}:{}&0.046672178229&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5}}&{}\approx{}&0.857603143946&{}:{}&-0.014737453291&{}:{}&0.157134309346&,\\ A^*_{\mathbf{5}}&{}\approx{}&0.000000000000&{}:{}&-0.103495636770&{}:{}&1.103495636770&,\\B^*_{\mathbf{5}}&{}\approx{}&0.845147817462&{}:{}&0.000000000000&{}:{}&0.154852182538&,\\C^*_{\mathbf{5}}&{}\approx{}&1.017484936752&{}:{}&-0.017484936752&{}:{}&0.000000000000&. \end{alignedat} \]
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Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5}}&{}\approx{}&0.899286975663&{}:{}&0.014963046273&{}:{}&0.085749978064&,\\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.837625355554&{}:{}&0.006840662084&{}:{}&0.155533982361&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.963301984604&{}:{}&0.032655592164&{}:{}&0.004042423232&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.953151902146&{}:{}&0.000175919625&{}:{}&0.046672178229&,\\ A^*_{\mathbf{5}}&{}\approx{}&0.000000000000&{}:{}&-0.103495636770&{}:{}&1.103495636770&,\\B^*_{\mathbf{5}}&{}\approx{}&0.845147817462&{}:{}&0.000000000000&{}:{}&0.154852182538&,\\C^*_{\mathbf{5}}&{}\approx{}&1.017484936752&{}:{}&-0.017484936752&{}:{}&0.000000000000&. \end{alignedat} \]
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