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,
\[ \begin{alignedat}{4} I&{}\approx{}&0.200000000000&{}:{}&0.333333333333&{}:{}&0.466666666667&, \\ P_{\mathbf{7}}&{}\approx{}&0.846486737601&{}:{}&0.150947939045&{}:{}&0.002565323354&, \\ P^-_{\mathbf{7}}&{}\approx{}&-1.481699780813&{}:{}&0.807770718910&{}:{}&1.673929061903&, \\ P^+_{\mathbf{7}}&{}\approx{}&0.471129035537&{}:{}&0.256843015477&{}:{}&0.272027948987&, \\ Q_{\mathbf{7}}&{}\approx{}&0.577466102496&{}:{}&0.327164568049&{}:{}&0.095369329454&, \\ I^\prime_{\mathbf{7}}&{}\approx{}&0.613777132768&{}:{}&0.334609751724&{}:{}&0.051613115508&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{7}}\)
\(P^-_{\mathbf{7}}\)
\(P^+_{\mathbf{7}}\)
\(Q_{\mathbf{7}}\)
\(I^\prime_{\mathbf{7}}\)
7 (222)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7}}&{}\approx{}&0.339626576009&{}:{}&0.275155593329&{}:{}&0.385217830661&,\\B^\prime_{\mathbf{7}}&{}\approx{}&0.452113991507&{}:{}&-0.507046638357&{}:{}&1.054932646850&,\\C^\prime_{\mathbf{7}}&{}\approx{}&2.918870481961&{}:{}&4.864784136602&{}:{}&-6.783654618563&, \\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.782953582064&{}:{}&0.213419407240&{}:{}&0.003627010697&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.477700838466&{}:{}&0.520851463581&{}:{}&0.001447697953&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.369715208310&{}:{}&0.065928674661&{}:{}&0.564356117029&, \\ A^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&-0.408902602683&{}:{}&0.458584989628&{}:{}&0.950317613055&,\\B^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&1.469647391994&{}:{}&1.190665656617&{}:{}&-1.660313048611&,\\C^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&0.629297132966&{}:{}&-0.343070711143&{}:{}&0.713773578177&, \\ A^*_{\mathbf{7}}&{}\approx{}&0.000000000000&{}:{}&0.774291885177&{}:{}&0.225708114823&,\\B^*_{\mathbf{7}}&{}\approx{}&0.858257569496&{}:{}&0.000000000000&{}:{}&0.141742430504&,\\C^*_{\mathbf{7}}&{}\approx{}&0.638344598849&{}:{}&0.361655401151&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7}}}{B^\prime_{\mathbf{7}}}{C^\prime_{\mathbf{7}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7}}}{B^{\prime\prime}_{\mathbf{7}}}{C^{\prime\prime}_{\mathbf{7}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7}}}{B^{\prime\prime\prime}_{\mathbf{7}}}{C^{\prime\prime\prime}_{\mathbf{7}}}\)
\(\triangle{A^*_{\mathbf{7}}}{B^*_{\mathbf{7}}}{C^*_{\mathbf{7}}}\)
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Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7}}}}&{}\approx{}&0.825466779988&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7}}}}&{}\approx{}&2.260569957535&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7}}}}&{}\approx{}&14.594352409805&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.200000000000&{}:{}&0.333333333333&{}:{}&0.466666666667&,\\ A^\prime_{\mathbf{7}}&{}\approx{}&0.339626576009&{}:{}&0.275155593329&{}:{}&0.385217830661&,\\B^\prime_{\mathbf{7}}&{}\approx{}&0.452113991507&{}:{}&-0.507046638357&{}:{}&1.054932646850&,\\C^\prime_{\mathbf{7}}&{}\approx{}&2.918870481961&{}:{}&4.864784136602&{}:{}&-6.783654618563&. \end{alignedat} \]
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First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7}}&{}\approx{}&0.846486737601&{}:{}&0.150947939045&{}:{}&0.002565323354&,\\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.782953582064&{}:{}&0.213419407240&{}:{}&0.003627010697&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.477700838466&{}:{}&0.520851463581&{}:{}&0.001447697953&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.369715208310&{}:{}&0.065928674661&{}:{}&0.564356117029&. \end{alignedat} \]
7 (222)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7}}&{}\approx{}&-1.481699780813&{}:{}&0.807770718910&{}:{}&1.673929061903&,\\ A^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&-0.408902602683&{}:{}&0.458584989628&{}:{}&0.950317613055&,\\B^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&1.469647391994&{}:{}&1.190665656617&{}:{}&-1.660313048611&,\\C^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&0.629297132966&{}:{}&-0.343070711143&{}:{}&0.713773578177&. \end{alignedat} \]
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Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7}}&{}\approx{}&0.471129035537&{}:{}&0.256843015477&{}:{}&0.272027948987&,\\ A^\prime_{\mathbf{7}}&{}\approx{}&0.339626576009&{}:{}&0.275155593329&{}:{}&0.385217830661&,\\B^\prime_{\mathbf{7}}&{}\approx{}&0.452113991507&{}:{}&-0.507046638357&{}:{}&1.054932646850&,\\C^\prime_{\mathbf{7}}&{}\approx{}&2.918870481961&{}:{}&4.864784136602&{}:{}&-6.783654618563&,\\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.782953582064&{}:{}&0.213419407240&{}:{}&0.003627010697&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.477700838466&{}:{}&0.520851463581&{}:{}&0.001447697953&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.369715208310&{}:{}&0.065928674661&{}:{}&0.564356117029&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7}}&{}\approx{}&0.577466102496&{}:{}&0.327164568049&{}:{}&0.095369329454&,\\ A^*_{\mathbf{7}}&{}\approx{}&0.000000000000&{}:{}&0.774291885177&{}:{}&0.225708114823&,\\B^*_{\mathbf{7}}&{}\approx{}&0.858257569496&{}:{}&0.000000000000&{}:{}&0.141742430504&,\\C^*_{\mathbf{7}}&{}\approx{}&0.638344598849&{}:{}&0.361655401151&{}:{}&0.000000000000&. \end{alignedat} \]
7 (222)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7}}&{}\approx{}&0.613777132768&{}:{}&0.334609751724&{}:{}&0.051613115508&,\\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.782953582064&{}:{}&0.213419407240&{}:{}&0.003627010697&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.477700838466&{}:{}&0.520851463581&{}:{}&0.001447697953&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.369715208310&{}:{}&0.065928674661&{}:{}&0.564356117029&,\\ A^*_{\mathbf{7}}&{}\approx{}&0.000000000000&{}:{}&0.774291885177&{}:{}&0.225708114823&,\\B^*_{\mathbf{7}}&{}\approx{}&0.858257569496&{}:{}&0.000000000000&{}:{}&0.141742430504&,\\C^*_{\mathbf{7}}&{}\approx{}&0.638344598849&{}:{}&0.361655401151&{}:{}&0.000000000000&. \end{alignedat} \]
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