Derousseau's Generalization of the Malfatti circles

Isosceles Triangle with 90° Top Angle

\(A=45\degree\), \(B=45\degree\), \(C=90\degree\).

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Malfatti circles

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Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.292893218813&{}:{}&0.292893218813&{}:{}&0.414213562373&, \\ P_{\mathbf{6}}&{}\approx{}&0.499298017376&{}:{}&0.499298017376&{}:{}&0.001403965248&, \\ P^-_{\mathbf{6}}&{}\approx{}&0.705124058166&{}:{}&0.705124058166&{}:{}&-0.410248116331&, \\ P^+_{\mathbf{6}}&{}\approx{}&0.430560844709&{}:{}&0.430560844709&{}:{}&0.138878310581&, \\ Q_{\mathbf{6}}&{}\approx{}&0.521483068724&{}:{}&0.521483068724&{}:{}&-0.042966137447&, \\ I^\prime_{\mathbf{6}}&{}\approx{}&0.484716815781&{}:{}&0.484716815781&{}:{}&0.030566368437&, \end{alignedat} \]
\(I\) Incenter
\(P_{\mathbf{6}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{6}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{6}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{6}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{6}}\) Radical center of the Malfatti circles
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Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6}}&{}\approx{}&0.500000000000&{}:{}&0.207106781187&{}:{}&0.292893218813&,\\B^\prime_{\mathbf{6}}&{}\approx{}&0.207106781187&{}:{}&0.500000000000&{}:{}&0.292893218813&,\\C^\prime_{\mathbf{6}}&{}\approx{}&1.679580427103&{}:{}&1.679580427103&{}:{}&-2.359160854207&, \\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.368744214168&{}:{}&0.629485748530&{}:{}&0.001770037301&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.629485748530&{}:{}&0.368744214168&{}:{}&0.001770037301&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.353553390593&{}:{}&0.353553390593&{}:{}&0.292893218813&, \\ A^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&0.497611377573&{}:{}&1.201340136528&{}:{}&-0.698951514101&,\\B^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&1.201340136528&{}:{}&0.497611377573&{}:{}&-0.698951514101&,\\C^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&0.386729540170&{}:{}&0.386729540170&{}:{}&0.226540919661&, \\ A^*_{\mathbf{6}}&{}\approx{}&0.000000000000&{}:{}&1.089790213552&{}:{}&-0.089790213552&,\\B^*_{\mathbf{6}}&{}\approx{}&1.089790213552&{}:{}&0.000000000000&{}:{}&-0.089790213552&,\\C^*_{\mathbf{6}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6}}}{B^\prime_{\mathbf{6}}}{C^\prime_{\mathbf{6}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6}}}{B^{\prime\prime}_{\mathbf{6}}}{C^{\prime\prime}_{\mathbf{6}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6}}}{B^{\prime\prime\prime}_{\mathbf{6}}}{C^{\prime\prime\prime}_{\mathbf{6}}}\)
\(\triangle{A^*_{\mathbf{6}}}{B^*_{\mathbf{6}}}{C^*_{\mathbf{6}}}\)
6 (220)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6}}}}&{}\approx{}&0.707106781187&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6}}}}&{}\approx{}&0.707106781187&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6}}}}&{}\approx{}&5.734446273312&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.292893218813&{}:{}&0.292893218813&{}:{}&0.414213562373&,\\ A^\prime_{\mathbf{6}}&{}\approx{}&0.500000000000&{}:{}&0.207106781187&{}:{}&0.292893218813&,\\B^\prime_{\mathbf{6}}&{}\approx{}&0.207106781187&{}:{}&0.500000000000&{}:{}&0.292893218813&,\\C^\prime_{\mathbf{6}}&{}\approx{}&1.679580427103&{}:{}&1.679580427103&{}:{}&-2.359160854207&. \end{alignedat} \]
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First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6}}&{}\approx{}&0.499298017376&{}:{}&0.499298017376&{}:{}&0.001403965248&,\\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.368744214168&{}:{}&0.629485748530&{}:{}&0.001770037301&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.629485748530&{}:{}&0.368744214168&{}:{}&0.001770037301&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.353553390593&{}:{}&0.353553390593&{}:{}&0.292893218813&. \end{alignedat} \]
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First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6}}&{}\approx{}&0.705124058166&{}:{}&0.705124058166&{}:{}&-0.410248116331&,\\ A^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&0.497611377573&{}:{}&1.201340136528&{}:{}&-0.698951514101&,\\B^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&1.201340136528&{}:{}&0.497611377573&{}:{}&-0.698951514101&,\\C^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&0.386729540170&{}:{}&0.386729540170&{}:{}&0.226540919661&. \end{alignedat} \]
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Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6}}&{}\approx{}&0.430560844709&{}:{}&0.430560844709&{}:{}&0.138878310581&,\\ A^\prime_{\mathbf{6}}&{}\approx{}&0.500000000000&{}:{}&0.207106781187&{}:{}&0.292893218813&,\\B^\prime_{\mathbf{6}}&{}\approx{}&0.207106781187&{}:{}&0.500000000000&{}:{}&0.292893218813&,\\C^\prime_{\mathbf{6}}&{}\approx{}&1.679580427103&{}:{}&1.679580427103&{}:{}&-2.359160854207&,\\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.368744214168&{}:{}&0.629485748530&{}:{}&0.001770037301&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.629485748530&{}:{}&0.368744214168&{}:{}&0.001770037301&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.353553390593&{}:{}&0.353553390593&{}:{}&0.292893218813&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6}}&{}\approx{}&0.521483068724&{}:{}&0.521483068724&{}:{}&-0.042966137447&,\\ A^*_{\mathbf{6}}&{}\approx{}&0.000000000000&{}:{}&1.089790213552&{}:{}&-0.089790213552&,\\B^*_{\mathbf{6}}&{}\approx{}&1.089790213552&{}:{}&0.000000000000&{}:{}&-0.089790213552&,\\C^*_{\mathbf{6}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
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Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6}}&{}\approx{}&0.484716815781&{}:{}&0.484716815781&{}:{}&0.030566368437&,\\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.368744214168&{}:{}&0.629485748530&{}:{}&0.001770037301&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.629485748530&{}:{}&0.368744214168&{}:{}&0.001770037301&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.353553390593&{}:{}&0.353553390593&{}:{}&0.292893218813&,\\ A^*_{\mathbf{6}}&{}\approx{}&0.000000000000&{}:{}&1.089790213552&{}:{}&-0.089790213552&,\\B^*_{\mathbf{6}}&{}\approx{}&1.089790213552&{}:{}&0.000000000000&{}:{}&-0.089790213552&,\\C^*_{\mathbf{6}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
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