Derousseau's Generalization of the Malfatti circles

Pythagorean Triangle U

\(C=90\degree\).   \(a:b:c=5:12:13\).


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7(222)

Malfatti circles

7 (222)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.166666666667&{}:{}&0.400000000000&{}:{}&0.433333333333&, \\ P_{\mathbf{7}}&{}\approx{}&0.958298871449&{}:{}&0.030748359751&{}:{}&0.010952768800&, \\ P^-_{\mathbf{7}}&{}\approx{}&-5.700464752738&{}:{}&3.136684898725&{}:{}&3.563779854013&, \\ P^+_{\mathbf{7}}&{}\approx{}&0.537467329224&{}:{}&0.227042467414&{}:{}&0.235490203362&, \\ Q_{\mathbf{7}}&{}\approx{}&0.638531483916&{}:{}&0.208824849399&{}:{}&0.152643666685&, \\ I^\prime_{\mathbf{7}}&{}\approx{}&0.689709175272&{}:{}&0.191395671947&{}:{}&0.118895152781&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{7}}\)
\(P^-_{\mathbf{7}}\)
\(P^+_{\mathbf{7}}\)
\(Q_{\mathbf{7}}\)
\(I^\prime_{\mathbf{7}}\)
7 (222)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7}}&{}\approx{}&0.448502526124&{}:{}&0.264718787460&{}:{}&0.286778686416&,\\B^\prime_{\mathbf{7}}&{}\approx{}&0.733914302124&{}:{}&-1.642091487647&{}:{}&1.908177185523&,\\C^\prime_{\mathbf{7}}&{}\approx{}&1.165631799079&{}:{}&2.797516317790&{}:{}&-2.963148116870&, \\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.900713962997&{}:{}&0.073208637035&{}:{}&0.026077399969&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.510495673307&{}:{}&0.483669674008&{}:{}&0.005834652685&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.485792576122&{}:{}&0.015587334327&{}:{}&0.498620089550&, \\ A^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&-0.264582514778&{}:{}&0.591988320762&{}:{}&0.672594194016&,\\B^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&1.036210882669&{}:{}&0.611600765739&{}:{}&-0.647811648408&,\\C^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&0.918136196316&{}:{}&-0.505205113421&{}:{}&0.587068917104&, \\ A^*_{\mathbf{7}}&{}\approx{}&0.000000000000&{}:{}&0.577712415072&{}:{}&0.422287584928&,\\B^*_{\mathbf{7}}&{}\approx{}&0.807067162602&{}:{}&0.000000000000&{}:{}&0.192932837398&,\\C^*_{\mathbf{7}}&{}\approx{}&0.753557221220&{}:{}&0.246442778780&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7}}}{B^\prime_{\mathbf{7}}}{C^\prime_{\mathbf{7}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7}}}{B^{\prime\prime}_{\mathbf{7}}}{C^{\prime\prime}_{\mathbf{7}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7}}}{B^{\prime\prime\prime}_{\mathbf{7}}}{C^{\prime\prime\prime}_{\mathbf{7}}}\)
\(\triangle{A^*_{\mathbf{7}}}{B^*_{\mathbf{7}}}{C^*_{\mathbf{7}}}\)
7 (222)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7}}}}&{}\approx{}&0.661796968651&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7}}}}&{}\approx{}&4.403485812745&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7}}}}&{}\approx{}&6.993790794476&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.166666666667&{}:{}&0.400000000000&{}:{}&0.433333333333&,\\ A^\prime_{\mathbf{7}}&{}\approx{}&0.448502526124&{}:{}&0.264718787460&{}:{}&0.286778686416&,\\B^\prime_{\mathbf{7}}&{}\approx{}&0.733914302124&{}:{}&-1.642091487647&{}:{}&1.908177185523&,\\C^\prime_{\mathbf{7}}&{}\approx{}&1.165631799079&{}:{}&2.797516317790&{}:{}&-2.963148116870&. \end{alignedat} \]
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First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7}}&{}\approx{}&0.958298871449&{}:{}&0.030748359751&{}:{}&0.010952768800&,\\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.900713962997&{}:{}&0.073208637035&{}:{}&0.026077399969&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.510495673307&{}:{}&0.483669674008&{}:{}&0.005834652685&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.485792576122&{}:{}&0.015587334327&{}:{}&0.498620089550&. \end{alignedat} \]
7 (222)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7}}&{}\approx{}&-5.700464752738&{}:{}&3.136684898725&{}:{}&3.563779854013&,\\ A^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&-0.264582514778&{}:{}&0.591988320762&{}:{}&0.672594194016&,\\B^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&1.036210882669&{}:{}&0.611600765739&{}:{}&-0.647811648408&,\\C^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&0.918136196316&{}:{}&-0.505205113421&{}:{}&0.587068917104&. \end{alignedat} \]
7 (222)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7}}&{}\approx{}&0.537467329224&{}:{}&0.227042467414&{}:{}&0.235490203362&,\\ A^\prime_{\mathbf{7}}&{}\approx{}&0.448502526124&{}:{}&0.264718787460&{}:{}&0.286778686416&,\\B^\prime_{\mathbf{7}}&{}\approx{}&0.733914302124&{}:{}&-1.642091487647&{}:{}&1.908177185523&,\\C^\prime_{\mathbf{7}}&{}\approx{}&1.165631799079&{}:{}&2.797516317790&{}:{}&-2.963148116870&,\\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.900713962997&{}:{}&0.073208637035&{}:{}&0.026077399969&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.510495673307&{}:{}&0.483669674008&{}:{}&0.005834652685&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.485792576122&{}:{}&0.015587334327&{}:{}&0.498620089550&, \end{alignedat} \]
7 (222)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7}}&{}\approx{}&0.638531483916&{}:{}&0.208824849399&{}:{}&0.152643666685&,\\ A^*_{\mathbf{7}}&{}\approx{}&0.000000000000&{}:{}&0.577712415072&{}:{}&0.422287584928&,\\B^*_{\mathbf{7}}&{}\approx{}&0.807067162602&{}:{}&0.000000000000&{}:{}&0.192932837398&,\\C^*_{\mathbf{7}}&{}\approx{}&0.753557221220&{}:{}&0.246442778780&{}:{}&0.000000000000&. \end{alignedat} \]
7 (222)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7}}&{}\approx{}&0.689709175272&{}:{}&0.191395671947&{}:{}&0.118895152781&,\\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.900713962997&{}:{}&0.073208637035&{}:{}&0.026077399969&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.510495673307&{}:{}&0.483669674008&{}:{}&0.005834652685&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.485792576122&{}:{}&0.015587334327&{}:{}&0.498620089550&,\\ A^*_{\mathbf{7}}&{}\approx{}&0.000000000000&{}:{}&0.577712415072&{}:{}&0.422287584928&,\\B^*_{\mathbf{7}}&{}\approx{}&0.807067162602&{}:{}&0.000000000000&{}:{}&0.192932837398&,\\C^*_{\mathbf{7}}&{}\approx{}&0.753557221220&{}:{}&0.246442778780&{}:{}&0.000000000000&. \end{alignedat} \]
7 (222)