Derousseau's Generalization of the Malfatti circles

Pythagorean Triangle U

\(C=90\degree\).   \(a:b:c=5:12:13\).


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6(220)

Malfatti circles

6 (220)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.166666666667&{}:{}&0.400000000000&{}:{}&0.433333333333&, \\ P_{\mathbf{6}}&{}\approx{}&0.968595378743&{}:{}&0.031078737590&{}:{}&0.000325883667&, \\ P^-_{\mathbf{6}}&{}\approx{}&1.165102327170&{}:{}&-0.059322803588&{}:{}&-0.105779523583&, \\ P^+_{\mathbf{6}}&{}\approx{}&0.836719117164&{}:{}&0.091747418198&{}:{}&0.071533464638&, \\ Q_{\mathbf{6}}&{}\approx{}&0.777590457810&{}:{}&0.254302590141&{}:{}&-0.031893047950&, \\ I^\prime_{\mathbf{6}}&{}\approx{}&0.765064993596&{}:{}&0.212307061849&{}:{}&0.022627944554&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{6}}\)
\(P^-_{\mathbf{6}}\)
\(P^+_{\mathbf{6}}\)
\(Q_{\mathbf{6}}\)
\(I^\prime_{\mathbf{6}}\)
6 (220)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6}}&{}\approx{}&0.850871453186&{}:{}&0.071581702471&{}:{}&0.077546844343&,\\B^\prime_{\mathbf{6}}&{}\approx{}&0.465668350749&{}:{}&-0.676406062698&{}:{}&1.210737711948&,\\C^\prime_{\mathbf{6}}&{}\approx{}&1.234667392808&{}:{}&2.963201742740&{}:{}&-3.197869135548&, \\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.676272942813&{}:{}&0.320367763033&{}:{}&0.003359294154&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.859924081571&{}:{}&0.139786597191&{}:{}&0.000289321237&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.827684584354&{}:{}&0.026557417647&{}:{}&0.145757997999&, \\ A^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&-0.328664738688&{}:{}&0.477401613156&{}:{}&0.851263125532&,\\B^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&1.006707068095&{}:{}&0.084691765782&{}:{}&-0.091398833877&,\\C^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&0.961333481241&{}:{}&-0.048947629715&{}:{}&0.087614148474&, \\ A^*_{\mathbf{6}}&{}\approx{}&0.000000000000&{}:{}&1.143397840021&{}:{}&-0.143397840021&,\\B^*_{\mathbf{6}}&{}\approx{}&1.042769422997&{}:{}&0.000000000000&{}:{}&-0.042769422997&,\\C^*_{\mathbf{6}}&{}\approx{}&0.753557221220&{}:{}&0.246442778780&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6}}}{B^\prime_{\mathbf{6}}}{C^\prime_{\mathbf{6}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6}}}{B^{\prime\prime}_{\mathbf{6}}}{C^{\prime\prime}_{\mathbf{6}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6}}}{B^{\prime\prime\prime}_{\mathbf{6}}}{C^{\prime\prime\prime}_{\mathbf{6}}}\)
\(\triangle{A^*_{\mathbf{6}}}{B^*_{\mathbf{6}}}{C^*_{\mathbf{6}}}\)
6 (220)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6}}}}&{}\approx{}&0.178954256177&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6}}}}&{}\approx{}&2.794010104496&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6}}}}&{}\approx{}&7.408004356849&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.166666666667&{}:{}&0.400000000000&{}:{}&0.433333333333&,\\ A^\prime_{\mathbf{6}}&{}\approx{}&0.850871453186&{}:{}&0.071581702471&{}:{}&0.077546844343&,\\B^\prime_{\mathbf{6}}&{}\approx{}&0.465668350749&{}:{}&-0.676406062698&{}:{}&1.210737711948&,\\C^\prime_{\mathbf{6}}&{}\approx{}&1.234667392808&{}:{}&2.963201742740&{}:{}&-3.197869135548&. \end{alignedat} \]
6 (220)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6}}&{}\approx{}&0.968595378743&{}:{}&0.031078737590&{}:{}&0.000325883667&,\\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.676272942813&{}:{}&0.320367763033&{}:{}&0.003359294154&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.859924081571&{}:{}&0.139786597191&{}:{}&0.000289321237&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.827684584354&{}:{}&0.026557417647&{}:{}&0.145757997999&. \end{alignedat} \]
6 (220)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6}}&{}\approx{}&1.165102327170&{}:{}&-0.059322803588&{}:{}&-0.105779523583&,\\ A^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&-0.328664738688&{}:{}&0.477401613156&{}:{}&0.851263125532&,\\B^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&1.006707068095&{}:{}&0.084691765782&{}:{}&-0.091398833877&,\\C^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&0.961333481241&{}:{}&-0.048947629715&{}:{}&0.087614148474&. \end{alignedat} \]
6 (220)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6}}&{}\approx{}&0.836719117164&{}:{}&0.091747418198&{}:{}&0.071533464638&,\\ A^\prime_{\mathbf{6}}&{}\approx{}&0.850871453186&{}:{}&0.071581702471&{}:{}&0.077546844343&,\\B^\prime_{\mathbf{6}}&{}\approx{}&0.465668350749&{}:{}&-0.676406062698&{}:{}&1.210737711948&,\\C^\prime_{\mathbf{6}}&{}\approx{}&1.234667392808&{}:{}&2.963201742740&{}:{}&-3.197869135548&,\\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.676272942813&{}:{}&0.320367763033&{}:{}&0.003359294154&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.859924081571&{}:{}&0.139786597191&{}:{}&0.000289321237&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.827684584354&{}:{}&0.026557417647&{}:{}&0.145757997999&, \end{alignedat} \]
6 (220)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6}}&{}\approx{}&0.777590457810&{}:{}&0.254302590141&{}:{}&-0.031893047950&,\\ A^*_{\mathbf{6}}&{}\approx{}&0.000000000000&{}:{}&1.143397840021&{}:{}&-0.143397840021&,\\B^*_{\mathbf{6}}&{}\approx{}&1.042769422997&{}:{}&0.000000000000&{}:{}&-0.042769422997&,\\C^*_{\mathbf{6}}&{}\approx{}&0.753557221220&{}:{}&0.246442778780&{}:{}&0.000000000000&. \end{alignedat} \]
6 (220)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6}}&{}\approx{}&0.765064993596&{}:{}&0.212307061849&{}:{}&0.022627944554&,\\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.676272942813&{}:{}&0.320367763033&{}:{}&0.003359294154&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.859924081571&{}:{}&0.139786597191&{}:{}&0.000289321237&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.827684584354&{}:{}&0.026557417647&{}:{}&0.145757997999&,\\ A^*_{\mathbf{6}}&{}\approx{}&0.000000000000&{}:{}&1.143397840021&{}:{}&-0.143397840021&,\\B^*_{\mathbf{6}}&{}\approx{}&1.042769422997&{}:{}&0.000000000000&{}:{}&-0.042769422997&,\\C^*_{\mathbf{6}}&{}\approx{}&0.753557221220&{}:{}&0.246442778780&{}:{}&0.000000000000&. \end{alignedat} \]
6 (220)