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

5 (202)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.292893218813&{}:{}&0.292893218813&{}:{}&0.414213562373&, \\ P_{\mathbf{5}}&{}\approx{}&0.911505099919&{}:{}&0.001426941760&{}:{}&0.087067958321&, \\ P^-_{\mathbf{5}}&{}\approx{}&1.076145164814&{}:{}&-0.076145164814&{}:{}&-0.000000000000&, \\ P^+_{\mathbf{5}}&{}\approx{}&0.804057944463&{}:{}&0.052051935131&{}:{}&0.143890120405&, \\ Q_{\mathbf{5}}&{}\approx{}&0.694130733431&{}:{}&-0.027464066764&{}:{}&0.333333333333&, \\ I^\prime_{\mathbf{5}}&{}\approx{}&0.710677371828&{}:{}&0.028118753208&{}:{}&0.261203874964&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{5}}\)
\(P^-_{\mathbf{5}}\)
\(P^+_{\mathbf{5}}\)
\(Q_{\mathbf{5}}\)
\(I^\prime_{\mathbf{5}}\)
5 (202)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5}}&{}\approx{}&0.851153302357&{}:{}&0.061654320878&{}:{}&0.087192376765&,\\B^\prime_{\mathbf{5}}&{}\approx{}&0.695704992003&{}:{}&-0.679580427103&{}:{}&0.983875435101&,\\C^\prime_{\mathbf{5}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&-0.000000000000&, \\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.581836654238&{}:{}&0.006742702007&{}:{}&0.411420643755&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.812606250236&{}:{}&0.109772717108&{}:{}&0.077621032656&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.838498727648&{}:{}&0.001312651844&{}:{}&0.160188620509&, \\ A^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&43.145659826952&{}:{}&-42.145659826952&{}:{}&-0.000000000000&,\\B^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&0.932456391348&{}:{}&0.067543608652&{}:{}&-0.000000000000&,\\C^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&0.969290044293&{}:{}&-0.068584381168&{}:{}&0.099294336875&, \\ A^*_{\mathbf{5}}&{}\approx{}&0.000000000000&{}:{}&-0.089790213552&{}:{}&1.089790213552&,\\B^*_{\mathbf{5}}&{}\approx{}&0.675576651179&{}:{}&0.000000000000&{}:{}&0.324423348821&,\\C^*_{\mathbf{5}}&{}\approx{}&1.041196100146&{}:{}&-0.041196100146&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5}}}{B^\prime_{\mathbf{5}}}{C^\prime_{\mathbf{5}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5}}}{B^{\prime\prime}_{\mathbf{5}}}{C^{\prime\prime}_{\mathbf{5}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5}}}{B^{\prime\prime\prime}_{\mathbf{5}}}{C^{\prime\prime\prime}_{\mathbf{5}}}\)
\(\triangle{A^*_{\mathbf{5}}}{B^*_{\mathbf{5}}}{C^*_{\mathbf{5}}}\)
5 (202)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5}}}}&{}\approx{}&0.210501018521&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5}}}}&{}\approx{}&2.375285419106&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5}}}}&{}\approx{}&1.707106781187&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.292893218813&{}:{}&0.292893218813&{}:{}&0.414213562373&,\\ A^\prime_{\mathbf{5}}&{}\approx{}&0.851153302357&{}:{}&0.061654320878&{}:{}&0.087192376765&,\\B^\prime_{\mathbf{5}}&{}\approx{}&0.695704992003&{}:{}&-0.679580427103&{}:{}&0.983875435101&,\\C^\prime_{\mathbf{5}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&-0.000000000000&. \end{alignedat} \]
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First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5}}&{}\approx{}&0.911505099919&{}:{}&0.001426941760&{}:{}&0.087067958321&,\\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.581836654238&{}:{}&0.006742702007&{}:{}&0.411420643755&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.812606250236&{}:{}&0.109772717108&{}:{}&0.077621032656&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.838498727648&{}:{}&0.001312651844&{}:{}&0.160188620509&. \end{alignedat} \]
5 (202)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5}}&{}\approx{}&1.076145164814&{}:{}&-0.076145164814&{}:{}&-0.000000000000&,\\ A^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&43.145659826952&{}:{}&-42.145659826952&{}:{}&-0.000000000000&,\\B^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&0.932456391348&{}:{}&0.067543608652&{}:{}&-0.000000000000&,\\C^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&0.969290044293&{}:{}&-0.068584381168&{}:{}&0.099294336875&. \end{alignedat} \]
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Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5}}&{}\approx{}&0.804057944463&{}:{}&0.052051935131&{}:{}&0.143890120405&,\\ A^\prime_{\mathbf{5}}&{}\approx{}&0.851153302357&{}:{}&0.061654320878&{}:{}&0.087192376765&,\\B^\prime_{\mathbf{5}}&{}\approx{}&0.695704992003&{}:{}&-0.679580427103&{}:{}&0.983875435101&,\\C^\prime_{\mathbf{5}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&-0.000000000000&,\\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.581836654238&{}:{}&0.006742702007&{}:{}&0.411420643755&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.812606250236&{}:{}&0.109772717108&{}:{}&0.077621032656&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.838498727648&{}:{}&0.001312651844&{}:{}&0.160188620509&, \end{alignedat} \]
5 (202)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5}}&{}\approx{}&0.694130733431&{}:{}&-0.027464066764&{}:{}&0.333333333333&,\\ A^*_{\mathbf{5}}&{}\approx{}&0.000000000000&{}:{}&-0.089790213552&{}:{}&1.089790213552&,\\B^*_{\mathbf{5}}&{}\approx{}&0.675576651179&{}:{}&0.000000000000&{}:{}&0.324423348821&,\\C^*_{\mathbf{5}}&{}\approx{}&1.041196100146&{}:{}&-0.041196100146&{}:{}&0.000000000000&. \end{alignedat} \]
5 (202)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5}}&{}\approx{}&0.710677371828&{}:{}&0.028118753208&{}:{}&0.261203874964&,\\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.581836654238&{}:{}&0.006742702007&{}:{}&0.411420643755&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.812606250236&{}:{}&0.109772717108&{}:{}&0.077621032656&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.838498727648&{}:{}&0.001312651844&{}:{}&0.160188620509&,\\ A^*_{\mathbf{5}}&{}\approx{}&0.000000000000&{}:{}&-0.089790213552&{}:{}&1.089790213552&,\\B^*_{\mathbf{5}}&{}\approx{}&0.675576651179&{}:{}&0.000000000000&{}:{}&0.324423348821&,\\C^*_{\mathbf{5}}&{}\approx{}&1.041196100146&{}:{}&-0.041196100146&{}:{}&0.000000000000&. \end{alignedat} \]
5 (202)