Derousseau's Generalization of the Malfatti circles

Pythagorean Triangle U

\(C=90\degree\).   \(a:b:c=5:12:13\).


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6c(330)

Malfatti circles

6c (330)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.250000000000&{}:{}&3.000000000000&{}:{}&-3.250000000000&, \\ P_{\mathbf{6c}}&{}\approx{}&0.121285650836&{}:{}&0.948641852015&{}:{}&-0.069927502851&, \\ P^-_{\mathbf{6c}}&{}\approx{}&0.060246789953&{}:{}&0.837708058440&{}:{}&0.102045151607&, \\ P^+_{\mathbf{6c}}&{}\approx{}&0.176367098613&{}:{}&1.048748470474&{}:{}&-0.225115569087&, \\ Q_{\mathbf{6c}}&{}\approx{}&0.348263202803&{}:{}&0.533475704939&{}:{}&0.118261092258&, \\ I^\prime_{\mathbf{6c}}&{}\approx{}&0.243410668045&{}:{}&1.054609448743&{}:{}&-0.298020116789&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{6c}}\)
\(P^-_{\mathbf{6c}}\)
\(P^+_{\mathbf{6c}}\)
\(Q_{\mathbf{6c}}\)
\(I^\prime_{\mathbf{6c}}\)
6c (330)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6c}}&{}\approx{}&1.289329219672&{}:{}&3.471950636069&{}:{}&-3.761279855741&,\\B^\prime_{\mathbf{6c}}&{}\approx{}&0.092656599941&{}:{}&1.148250559905&{}:{}&-0.240907159846&,\\C^\prime_{\mathbf{6c}}&{}\approx{}&0.203682854896&{}:{}&0.488838851750&{}:{}&0.307478293354&, \\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.127371232503&{}:{}&0.942071983811&{}:{}&-0.069443216314&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.337672839467&{}:{}&0.857013162395&{}:{}&-0.194686001862&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.164688966485&{}:{}&1.288123080481&{}:{}&-0.452812046966&, \\ A^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.067104488831&{}:{}&0.831595017749&{}:{}&0.101300493420&,\\B^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.185645154063&{}:{}&0.499911737745&{}:{}&0.314443108193&,\\C^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.083421125218&{}:{}&1.159938129376&{}:{}&-0.243359254594&, \\ A^*_{\mathbf{6c}}&{}\approx{}&0.000000000000&{}:{}&0.818544705828&{}:{}&0.181455294172&,\\B^*_{\mathbf{6c}}&{}\approx{}&0.746506037284&{}:{}&0.000000000000&{}:{}&0.253493962716&,\\C^*_{\mathbf{6c}}&{}\approx{}&0.394973160133&{}:{}&0.605026839867&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6c}}}{B^\prime_{\mathbf{6c}}}{C^\prime_{\mathbf{6c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6c}}}{B^{\prime\prime}_{\mathbf{6c}}}{C^{\prime\prime}_{\mathbf{6c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6c}}}{B^{\prime\prime\prime}_{\mathbf{6c}}}{C^{\prime\prime\prime}_{\mathbf{6c}}}\)
\(\triangle{A^*_{\mathbf{6c}}}{B^*_{\mathbf{6c}}}{C^*_{\mathbf{6c}}}\)
6c (330)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6c}}}}&{}\approx{}&1.157316878690&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6c}}}}&{}\approx{}&0.074125279953&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6c}}}}&{}\approx{}&0.162946283917&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.250000000000&{}:{}&3.000000000000&{}:{}&-3.250000000000&,\\ A^\prime_{\mathbf{6c}}&{}\approx{}&1.289329219672&{}:{}&3.471950636069&{}:{}&-3.761279855741&,\\B^\prime_{\mathbf{6c}}&{}\approx{}&0.092656599941&{}:{}&1.148250559905&{}:{}&-0.240907159846&,\\C^\prime_{\mathbf{6c}}&{}\approx{}&0.203682854896&{}:{}&0.488838851750&{}:{}&0.307478293354&. \end{alignedat} \]
6c (330)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6c}}&{}\approx{}&0.121285650836&{}:{}&0.948641852015&{}:{}&-0.069927502851&,\\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.127371232503&{}:{}&0.942071983811&{}:{}&-0.069443216314&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.337672839467&{}:{}&0.857013162395&{}:{}&-0.194686001862&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.164688966485&{}:{}&1.288123080481&{}:{}&-0.452812046966&. \end{alignedat} \]
6c (330)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6c}}&{}\approx{}&0.060246789953&{}:{}&0.837708058440&{}:{}&0.102045151607&,\\ A^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.067104488831&{}:{}&0.831595017749&{}:{}&0.101300493420&,\\B^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.185645154063&{}:{}&0.499911737745&{}:{}&0.314443108193&,\\C^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.083421125218&{}:{}&1.159938129376&{}:{}&-0.243359254594&. \end{alignedat} \]
6c (330)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6c}}&{}\approx{}&0.176367098613&{}:{}&1.048748470474&{}:{}&-0.225115569087&,\\ A^\prime_{\mathbf{6c}}&{}\approx{}&1.289329219672&{}:{}&3.471950636069&{}:{}&-3.761279855741&,\\B^\prime_{\mathbf{6c}}&{}\approx{}&0.092656599941&{}:{}&1.148250559905&{}:{}&-0.240907159846&,\\C^\prime_{\mathbf{6c}}&{}\approx{}&0.203682854896&{}:{}&0.488838851750&{}:{}&0.307478293354&,\\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.127371232503&{}:{}&0.942071983811&{}:{}&-0.069443216314&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.337672839467&{}:{}&0.857013162395&{}:{}&-0.194686001862&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.164688966485&{}:{}&1.288123080481&{}:{}&-0.452812046966&, \end{alignedat} \]
6c (330)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6c}}&{}\approx{}&0.348263202803&{}:{}&0.533475704939&{}:{}&0.118261092258&,\\ A^*_{\mathbf{6c}}&{}\approx{}&0.000000000000&{}:{}&0.818544705828&{}:{}&0.181455294172&,\\B^*_{\mathbf{6c}}&{}\approx{}&0.746506037284&{}:{}&0.000000000000&{}:{}&0.253493962716&,\\C^*_{\mathbf{6c}}&{}\approx{}&0.394973160133&{}:{}&0.605026839867&{}:{}&0.000000000000&. \end{alignedat} \]
6c (330)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6c}}&{}\approx{}&0.243410668045&{}:{}&1.054609448743&{}:{}&-0.298020116789&,\\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.127371232503&{}:{}&0.942071983811&{}:{}&-0.069443216314&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.337672839467&{}:{}&0.857013162395&{}:{}&-0.194686001862&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.164688966485&{}:{}&1.288123080481&{}:{}&-0.452812046966&,\\ A^*_{\mathbf{6c}}&{}\approx{}&0.000000000000&{}:{}&0.818544705828&{}:{}&0.181455294172&,\\B^*_{\mathbf{6c}}&{}\approx{}&0.746506037284&{}:{}&0.000000000000&{}:{}&0.253493962716&,\\C^*_{\mathbf{6c}}&{}\approx{}&0.394973160133&{}:{}&0.605026839867&{}:{}&0.000000000000&. \end{alignedat} \]
6c (330)