Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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Malfatti circles

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Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.200000000000&{}:{}&0.333333333333&{}:{}&0.466666666667&, \\ P&{}\approx{}&0.146960670558&{}:{}&0.254360157408&{}:{}&0.598679172034&, \\ P^-&{}\approx{}&0.129318325634&{}:{}&0.228091499336&{}:{}&0.642590175029&, \\ P^+&{}\approx{}&0.157555049665&{}:{}&0.270134711381&{}:{}&0.572310238954&, \\ Q&{}\approx{}&0.113377225166&{}:{}&0.200117955067&{}:{}&0.686504819768&, \\ I^\prime&{}\approx{}&0.172965088465&{}:{}&0.293769570429&{}:{}&0.533265341105&, \end{alignedat} \]
\(I\)
\(P\)
\(P^-\)
\(P^+\)
\(Q\)
\(I^\prime\)
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Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.327040090657&{}:{}&0.280399962226&{}:{}&0.392559947117&,\\B^\prime&{}\approx{}&0.147886008493&{}:{}&0.507046638357&{}:{}&0.345067353150&,\\C^\prime&{}\approx{}&0.081129518039&{}:{}&0.135215863398&{}:{}&0.783654618563&, \\ A^{\prime\prime}&{}\approx{}&0.104778228581&{}:{}&0.266938161975&{}:{}&0.628283609445&,\\B^{\prime\prime}&{}\approx{}&0.161133715053&{}:{}&0.182449852251&{}:{}&0.656416432696&,\\C^{\prime\prime}&{}\approx{}&0.231695575894&{}:{}&0.401019694120&{}:{}&0.367284729986&, \\ A^{\prime\prime\prime}&{}\approx{}&0.070982724666&{}:{}&0.243373611136&{}:{}&0.685643664198&,\\B^{\prime\prime\prime}&{}\approx{}&0.146489143169&{}:{}&0.125597904919&{}:{}&0.727912951912&,\\C^{\prime\prime\prime}&{}\approx{}&0.252261479537&{}:{}&0.444938478830&{}:{}&0.302800041633&, \\ A^*&{}\approx{}&0.000000000000&{}:{}&0.225708114823&{}:{}&0.774291885177&,\\B^*&{}\approx{}&0.141742430504&{}:{}&0.000000000000&{}:{}&0.858257569496&,\\C^*&{}\approx{}&0.361655401151&{}:{}&0.638344598849&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime}{B^\prime}{C^\prime}\)
\(\triangle{A^{\prime\prime}}{B^{\prime\prime}}{C^{\prime\prime}}\)
\(\triangle{A^{\prime\prime\prime}}{B^{\prime\prime\prime}}{C^{\prime\prime\prime}}\)
\(\triangle{A^*}{B^*}{C^*}\)
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Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime}}&{}\approx{}&0.841199886679&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime}}&{}\approx{}&0.739430042465&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime}}&{}\approx{}&0.405647590195&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.200000000000&{}:{}&0.333333333333&{}:{}&0.466666666667&,\\ A^\prime&{}\approx{}&0.327040090657&{}:{}&0.280399962226&{}:{}&0.392559947117&,\\B^\prime&{}\approx{}&0.147886008493&{}:{}&0.507046638357&{}:{}&0.345067353150&,\\C^\prime&{}\approx{}&0.081129518039&{}:{}&0.135215863398&{}:{}&0.783654618563&. \end{alignedat} \]
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First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P&{}\approx{}&0.146960670558&{}:{}&0.254360157408&{}:{}&0.598679172034&,\\ A^{\prime\prime}&{}\approx{}&0.104778228581&{}:{}&0.266938161975&{}:{}&0.628283609445&,\\B^{\prime\prime}&{}\approx{}&0.161133715053&{}:{}&0.182449852251&{}:{}&0.656416432696&,\\C^{\prime\prime}&{}\approx{}&0.231695575894&{}:{}&0.401019694120&{}:{}&0.367284729986&. \end{alignedat} \]
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First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-&{}\approx{}&0.129318325634&{}:{}&0.228091499336&{}:{}&0.642590175029&,\\ A^{\prime\prime\prime}&{}\approx{}&0.070982724666&{}:{}&0.243373611136&{}:{}&0.685643664198&,\\B^{\prime\prime\prime}&{}\approx{}&0.146489143169&{}:{}&0.125597904919&{}:{}&0.727912951912&,\\C^{\prime\prime\prime}&{}\approx{}&0.252261479537&{}:{}&0.444938478830&{}:{}&0.302800041633&. \end{alignedat} \]
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Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+&{}\approx{}&0.157555049665&{}:{}&0.270134711381&{}:{}&0.572310238954&,\\ A^\prime&{}\approx{}&0.327040090657&{}:{}&0.280399962226&{}:{}&0.392559947117&,\\B^\prime&{}\approx{}&0.147886008493&{}:{}&0.507046638357&{}:{}&0.345067353150&,\\C^\prime&{}\approx{}&0.081129518039&{}:{}&0.135215863398&{}:{}&0.783654618563&,\\ A^{\prime\prime}&{}\approx{}&0.104778228581&{}:{}&0.266938161975&{}:{}&0.628283609445&,\\B^{\prime\prime}&{}\approx{}&0.161133715053&{}:{}&0.182449852251&{}:{}&0.656416432696&,\\C^{\prime\prime}&{}\approx{}&0.231695575894&{}:{}&0.401019694120&{}:{}&0.367284729986&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q&{}\approx{}&0.113377225166&{}:{}&0.200117955067&{}:{}&0.686504819768&,\\ A^*&{}\approx{}&0.000000000000&{}:{}&0.225708114823&{}:{}&0.774291885177&,\\B^*&{}\approx{}&0.141742430504&{}:{}&0.000000000000&{}:{}&0.858257569496&,\\C^*&{}\approx{}&0.361655401151&{}:{}&0.638344598849&{}:{}&0.000000000000&. \end{alignedat} \]
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Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.172965088465&{}:{}&0.293769570429&{}:{}&0.533265341105&,\\ A^{\prime\prime}&{}\approx{}&0.104778228581&{}:{}&0.266938161975&{}:{}&0.628283609445&,\\B^{\prime\prime}&{}\approx{}&0.161133715053&{}:{}&0.182449852251&{}:{}&0.656416432696&,\\C^{\prime\prime}&{}\approx{}&0.231695575894&{}:{}&0.401019694120&{}:{}&0.367284729986&,\\ A^*&{}\approx{}&0.000000000000&{}:{}&0.225708114823&{}:{}&0.774291885177&,\\B^*&{}\approx{}&0.141742430504&{}:{}&0.000000000000&{}:{}&0.858257569496&,\\C^*&{}\approx{}&0.361655401151&{}:{}&0.638344598849&{}:{}&0.000000000000&. \end{alignedat} \]
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