Derousseau's Generalization of the Malfatti circles

Half of an Equilateral Triangle

\(A=30\degree\), \(B=60\degree\), \(C=90\degree\).

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Malfatti circles

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Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.211324865405&{}:{}&0.366025403784&{}:{}&0.422649730810&, \\ P&{}\approx{}&0.179375592383&{}:{}&0.344844360463&{}:{}&0.475780047154&, \\ P^-&{}\approx{}&0.168037754561&{}:{}&0.337327844081&{}:{}&0.494634401359&, \\ P^+&{}\approx{}&0.186006915339&{}:{}&0.349240652633&{}:{}&0.464752432029&, \\ Q&{}\approx{}&0.161771971219&{}:{}&0.329250639680&{}:{}&0.508977389101&, \\ I^\prime&{}\approx{}&0.195007645337&{}:{}&0.355845798025&{}:{}&0.449146556638&, \end{alignedat} \]
\(I\) Incenter
\(P\) First Ajima-Malfatti point
\(P^-\) First Malfatti-Rabinowitz point
\(P^+\) Gergonne point of the Malfatti triangle
\(Q\) Second Malfatti-Rabinowitz point
\(I^\prime\) Radical center of the Malfatti circles
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Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.375155551113&{}:{}&0.289991317938&{}:{}&0.334853130949&,\\B^\prime&{}\approx{}&0.133366525832&{}:{}&0.599900422505&{}:{}&0.266733051664&,\\C^\prime&{}\approx{}&0.107206358682&{}:{}&0.185686860132&{}:{}&0.707106781187&, \\ A^{\prime\prime}&{}\approx{}&0.118864099197&{}:{}&0.370272616039&{}:{}&0.510863284764&,\\B^{\prime\prime}&{}\approx{}&0.211803640019&{}:{}&0.226403395303&{}:{}&0.561792964678&,\\C^{\prime\prime}&{}\approx{}&0.240571655265&{}:{}&0.462492011892&{}:{}&0.296936332843&, \\ A^{\prime\prime\prime}&{}\approx{}&0.082686398402&{}:{}&0.371934449273&{}:{}&0.545379152325&,\\B^{\prime\prime\prime}&{}\approx{}&0.212018011131&{}:{}&0.163887705492&{}:{}&0.624094283377&,\\C^{\prime\prime\prime}&{}\approx{}&0.256408669230&{}:{}&0.514728275328&{}:{}&0.228863055443&, \\ A^*&{}\approx{}&0.000000000000&{}:{}&0.392793641318&{}:{}&0.607206358682&,\\B^*&{}\approx{}&0.241180954897&{}:{}&0.000000000000&{}:{}&0.758819045103&,\\C^*&{}\approx{}&0.329459311299&{}:{}&0.670540688701&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime}{B^\prime}{C^\prime}\)
\(\triangle{A^{\prime\prime}}{B^{\prime\prime}}{C^{\prime\prime}}\)
\(\triangle{A^{\prime\prime\prime}}{B^{\prime\prime\prime}}{C^{\prime\prime\prime}}\)
\(\triangle{A^*}{B^*}{C^*}\)
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Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime}}&{}\approx{}&0.792271014361&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime}}&{}\approx{}&0.631097176265&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime}}&{}\approx{}&0.507305936177&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.211324865405&{}:{}&0.366025403784&{}:{}&0.422649730810&,\\ A^\prime&{}\approx{}&0.375155551113&{}:{}&0.289991317938&{}:{}&0.334853130949&,\\B^\prime&{}\approx{}&0.133366525832&{}:{}&0.599900422505&{}:{}&0.266733051664&,\\C^\prime&{}\approx{}&0.107206358682&{}:{}&0.185686860132&{}:{}&0.707106781187&. \end{alignedat} \]
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First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P&{}\approx{}&0.179375592383&{}:{}&0.344844360463&{}:{}&0.475780047154&,\\ A^{\prime\prime}&{}\approx{}&0.118864099197&{}:{}&0.370272616039&{}:{}&0.510863284764&,\\B^{\prime\prime}&{}\approx{}&0.211803640019&{}:{}&0.226403395303&{}:{}&0.561792964678&,\\C^{\prime\prime}&{}\approx{}&0.240571655265&{}:{}&0.462492011892&{}:{}&0.296936332843&. \end{alignedat} \]
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First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-&{}\approx{}&0.168037754561&{}:{}&0.337327844081&{}:{}&0.494634401359&,\\ A^{\prime\prime\prime}&{}\approx{}&0.082686398402&{}:{}&0.371934449273&{}:{}&0.545379152325&,\\B^{\prime\prime\prime}&{}\approx{}&0.212018011131&{}:{}&0.163887705492&{}:{}&0.624094283377&,\\C^{\prime\prime\prime}&{}\approx{}&0.256408669230&{}:{}&0.514728275328&{}:{}&0.228863055443&. \end{alignedat} \]
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Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+&{}\approx{}&0.186006915339&{}:{}&0.349240652633&{}:{}&0.464752432029&,\\ A^\prime&{}\approx{}&0.375155551113&{}:{}&0.289991317938&{}:{}&0.334853130949&,\\B^\prime&{}\approx{}&0.133366525832&{}:{}&0.599900422505&{}:{}&0.266733051664&,\\C^\prime&{}\approx{}&0.107206358682&{}:{}&0.185686860132&{}:{}&0.707106781187&,\\ A^{\prime\prime}&{}\approx{}&0.118864099197&{}:{}&0.370272616039&{}:{}&0.510863284764&,\\B^{\prime\prime}&{}\approx{}&0.211803640019&{}:{}&0.226403395303&{}:{}&0.561792964678&,\\C^{\prime\prime}&{}\approx{}&0.240571655265&{}:{}&0.462492011892&{}:{}&0.296936332843&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q&{}\approx{}&0.161771971219&{}:{}&0.329250639680&{}:{}&0.508977389101&,\\ A^*&{}\approx{}&0.000000000000&{}:{}&0.392793641318&{}:{}&0.607206358682&,\\B^*&{}\approx{}&0.241180954897&{}:{}&0.000000000000&{}:{}&0.758819045103&,\\C^*&{}\approx{}&0.329459311299&{}:{}&0.670540688701&{}:{}&0.000000000000&. \end{alignedat} \]
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Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.195007645337&{}:{}&0.355845798025&{}:{}&0.449146556638&,\\ A^{\prime\prime}&{}\approx{}&0.118864099197&{}:{}&0.370272616039&{}:{}&0.510863284764&,\\B^{\prime\prime}&{}\approx{}&0.211803640019&{}:{}&0.226403395303&{}:{}&0.561792964678&,\\C^{\prime\prime}&{}\approx{}&0.240571655265&{}:{}&0.462492011892&{}:{}&0.296936332843&,\\ A^*&{}\approx{}&0.000000000000&{}:{}&0.392793641318&{}:{}&0.607206358682&,\\B^*&{}\approx{}&0.241180954897&{}:{}&0.000000000000&{}:{}&0.758819045103&,\\C^*&{}\approx{}&0.329459311299&{}:{}&0.670540688701&{}:{}&0.000000000000&. \end{alignedat} \]
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