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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3(022)

Malfatti circles

3 (022)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.292893218813&{}:{}&0.292893218813&{}:{}&0.414213562373&, \\ P_{\mathbf{3}}&{}\approx{}&0.001426941760&{}:{}&0.911505099919&{}:{}&0.087067958321&, \\ P^-_{\mathbf{3}}&{}\approx{}&-0.076145164814&{}:{}&1.076145164814&{}:{}&-0.000000000000&, \\ P^+_{\mathbf{3}}&{}\approx{}&0.052051935131&{}:{}&0.804057944463&{}:{}&0.143890120405&, \\ Q_{\mathbf{3}}&{}\approx{}&-0.027464066764&{}:{}&0.694130733431&{}:{}&0.333333333333&, \\ I^\prime_{\mathbf{3}}&{}\approx{}&0.028118753208&{}:{}&0.710677371828&{}:{}&0.261203874964&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{3}}\)
\(P^-_{\mathbf{3}}\)
\(P^+_{\mathbf{3}}\)
\(Q_{\mathbf{3}}\)
\(I^\prime_{\mathbf{3}}\)
3 (022)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3}}&{}\approx{}&-0.679580427103&{}:{}&0.695704992003&{}:{}&0.983875435101&,\\B^\prime_{\mathbf{3}}&{}\approx{}&0.061654320878&{}:{}&0.851153302357&{}:{}&0.087192376765&,\\C^\prime_{\mathbf{3}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&-0.000000000000&, \\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.109772717108&{}:{}&0.812606250236&{}:{}&0.077621032656&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.006742702007&{}:{}&0.581836654238&{}:{}&0.411420643755&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.001312651844&{}:{}&0.838498727648&{}:{}&0.160188620509&, \\ A^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&0.067543608652&{}:{}&0.932456391348&{}:{}&-0.000000000000&,\\B^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&-42.145659826952&{}:{}&43.145659826952&{}:{}&-0.000000000000&,\\C^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&-0.068584381168&{}:{}&0.969290044293&{}:{}&0.099294336875&, \\ A^*_{\mathbf{3}}&{}\approx{}&0.000000000000&{}:{}&0.675576651179&{}:{}&0.324423348821&,\\B^*_{\mathbf{3}}&{}\approx{}&-0.089790213552&{}:{}&0.000000000000&{}:{}&1.089790213552&,\\C^*_{\mathbf{3}}&{}\approx{}&-0.041196100146&{}:{}&1.041196100146&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3}}}{B^\prime_{\mathbf{3}}}{C^\prime_{\mathbf{3}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3}}}{B^{\prime\prime}_{\mathbf{3}}}{C^{\prime\prime}_{\mathbf{3}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3}}}{B^{\prime\prime\prime}_{\mathbf{3}}}{C^{\prime\prime\prime}_{\mathbf{3}}}\)
\(\triangle{A^*_{\mathbf{3}}}{B^*_{\mathbf{3}}}{C^*_{\mathbf{3}}}\)
3 (022)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3}}}}&{}\approx{}&2.375285419106&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3}}}}&{}\approx{}&0.210501018521&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3}}}}&{}\approx{}&1.707106781187&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.292893218813&{}:{}&0.292893218813&{}:{}&0.414213562373&,\\ A^\prime_{\mathbf{3}}&{}\approx{}&-0.679580427103&{}:{}&0.695704992003&{}:{}&0.983875435101&,\\B^\prime_{\mathbf{3}}&{}\approx{}&0.061654320878&{}:{}&0.851153302357&{}:{}&0.087192376765&,\\C^\prime_{\mathbf{3}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&-0.000000000000&. \end{alignedat} \]
3 (022)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3}}&{}\approx{}&0.001426941760&{}:{}&0.911505099919&{}:{}&0.087067958321&,\\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.109772717108&{}:{}&0.812606250236&{}:{}&0.077621032656&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.006742702007&{}:{}&0.581836654238&{}:{}&0.411420643755&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.001312651844&{}:{}&0.838498727648&{}:{}&0.160188620509&. \end{alignedat} \]
3 (022)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3}}&{}\approx{}&-0.076145164814&{}:{}&1.076145164814&{}:{}&-0.000000000000&,\\ A^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&0.067543608652&{}:{}&0.932456391348&{}:{}&-0.000000000000&,\\B^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&-42.145659826952&{}:{}&43.145659826952&{}:{}&-0.000000000000&,\\C^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&-0.068584381168&{}:{}&0.969290044293&{}:{}&0.099294336875&. \end{alignedat} \]
3 (022)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3}}&{}\approx{}&0.052051935131&{}:{}&0.804057944463&{}:{}&0.143890120405&,\\ A^\prime_{\mathbf{3}}&{}\approx{}&-0.679580427103&{}:{}&0.695704992003&{}:{}&0.983875435101&,\\B^\prime_{\mathbf{3}}&{}\approx{}&0.061654320878&{}:{}&0.851153302357&{}:{}&0.087192376765&,\\C^\prime_{\mathbf{3}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&-0.000000000000&,\\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.109772717108&{}:{}&0.812606250236&{}:{}&0.077621032656&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.006742702007&{}:{}&0.581836654238&{}:{}&0.411420643755&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.001312651844&{}:{}&0.838498727648&{}:{}&0.160188620509&, \end{alignedat} \]
3 (022)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3}}&{}\approx{}&-0.027464066764&{}:{}&0.694130733431&{}:{}&0.333333333333&,\\ A^*_{\mathbf{3}}&{}\approx{}&0.000000000000&{}:{}&0.675576651179&{}:{}&0.324423348821&,\\B^*_{\mathbf{3}}&{}\approx{}&-0.089790213552&{}:{}&0.000000000000&{}:{}&1.089790213552&,\\C^*_{\mathbf{3}}&{}\approx{}&-0.041196100146&{}:{}&1.041196100146&{}:{}&0.000000000000&. \end{alignedat} \]
3 (022)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3}}&{}\approx{}&0.028118753208&{}:{}&0.710677371828&{}:{}&0.261203874964&,\\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.109772717108&{}:{}&0.812606250236&{}:{}&0.077621032656&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.006742702007&{}:{}&0.581836654238&{}:{}&0.411420643755&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.001312651844&{}:{}&0.838498727648&{}:{}&0.160188620509&,\\ A^*_{\mathbf{3}}&{}\approx{}&0.000000000000&{}:{}&0.675576651179&{}:{}&0.324423348821&,\\B^*_{\mathbf{3}}&{}\approx{}&-0.089790213552&{}:{}&0.000000000000&{}:{}&1.089790213552&,\\C^*_{\mathbf{3}}&{}\approx{}&-0.041196100146&{}:{}&1.041196100146&{}:{}&0.000000000000&. \end{alignedat} \]
3 (022)