Derousseau's Generalization of the Malfatti circles

Isosceles Triangle with 135° Top Angle

\(A=22.5\degree\), \(B=22.5\degree\), \(C=135\degree\).

[Top] > Isosceles Triangle with 135° Top Angle > 2c (130)

2c (130)

Malfatti circles

2c (130)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&6.568535592272&{}:{}&6.568535592272&{}:{}&-12.137071184544&, \\ P_{\mathbf{2c}}&{}\approx{}&1.208535181249&{}:{}&0.548216543556&{}:{}&-0.756751724805&, \\ P^-_{\mathbf{2c}}&{}\approx{}&0.957080395984&{}:{}&0.265784096255&{}:{}&-0.222864492240&, \\ P^+_{\mathbf{2c}}&{}\approx{}&1.438420640281&{}:{}&0.806422458090&{}:{}&-1.244843098371&, \\ Q_{\mathbf{2c}}&{}\approx{}&1.143057852177&{}:{}&-0.769865097862&{}:{}&0.626807245685&, \\ I^\prime_{\mathbf{2c}}&{}\approx{}&1.672613439834&{}:{}&1.126528029260&{}:{}&-1.799141469094&, \end{alignedat} \]
\(I_{\mathbf{c}}\) Incenter
\(P_{\mathbf{2c}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{2c}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{2c}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{2c}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{2c}}\) Radical center of the Malfatti circles
2c (130)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{2c}}&{}\approx{}&1.375426647384&{}:{}&0.442845925067&{}:{}&-0.818272572451&,\\B^\prime_{\mathbf{2c}}&{}\approx{}&67.781251278963&{}:{}&58.462170210314&{}:{}&-125.243421489277&,\\C^\prime_{\mathbf{2c}}&{}\approx{}&0.783227248675&{}:{}&0.783227248675&{}:{}&-0.566454497351&, \\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.548560540122&{}:{}&1.442106609715&{}:{}&-1.990667149838&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.161529878283&{}:{}&0.565788409543&{}:{}&-0.727318287826&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.806470076964&{}:{}&0.819452174001&{}:{}&-1.625922250965&, \\ A^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.877746531603&{}:{}&0.757067273970&{}:{}&-0.634813805573&,\\B^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.918179918055&{}:{}&0.295626259650&{}:{}&-0.213806177705&,\\C^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.464600020884&{}:{}&0.406723818145&{}:{}&-0.871323839029&, \\ A^*_{\mathbf{2c}}&{}\approx{}&0.000000000000&{}:{}&5.381494871788&{}:{}&-4.381494871788&,\\B^*_{\mathbf{2c}}&{}\approx{}&0.645844620338&{}:{}&0.000000000000&{}:{}&0.354155379662&,\\C^*_{\mathbf{2c}}&{}\approx{}&3.062915447742&{}:{}&-2.062915447742&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{2c}}}{B^\prime_{\mathbf{2c}}}{C^\prime_{\mathbf{2c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{2c}}}{B^{\prime\prime}_{\mathbf{2c}}}{C^{\prime\prime}_{\mathbf{2c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{2c}}}{B^{\prime\prime\prime}_{\mathbf{2c}}}{C^{\prime\prime\prime}_{\mathbf{2c}}}\)
\(\triangle{A^*_{\mathbf{2c}}}{B^*_{\mathbf{2c}}}{C^*_{\mathbf{2c}}}\)
2c (130)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{2c}}}}&{}\approx{}&0.067419277683&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{2c}}}}&{}\approx{}&10.319081068649&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{2c}}}}&{}\approx{}&0.119239248638&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&6.568535592272&{}:{}&6.568535592272&{}:{}&-12.137071184544&,\\ A^\prime_{\mathbf{2c}}&{}\approx{}&1.375426647384&{}:{}&0.442845925067&{}:{}&-0.818272572451&,\\B^\prime_{\mathbf{2c}}&{}\approx{}&67.781251278963&{}:{}&58.462170210314&{}:{}&-125.243421489277&,\\C^\prime_{\mathbf{2c}}&{}\approx{}&0.783227248675&{}:{}&0.783227248675&{}:{}&-0.566454497351&. \end{alignedat} \]
2c (130)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{2c}}&{}\approx{}&1.208535181249&{}:{}&0.548216543556&{}:{}&-0.756751724805&,\\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.548560540122&{}:{}&1.442106609715&{}:{}&-1.990667149838&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.161529878283&{}:{}&0.565788409543&{}:{}&-0.727318287826&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.806470076964&{}:{}&0.819452174001&{}:{}&-1.625922250965&. \end{alignedat} \]
2c (130)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{2c}}&{}\approx{}&0.957080395984&{}:{}&0.265784096255&{}:{}&-0.222864492240&,\\ A^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.877746531603&{}:{}&0.757067273970&{}:{}&-0.634813805573&,\\B^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.918179918055&{}:{}&0.295626259650&{}:{}&-0.213806177705&,\\C^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.464600020884&{}:{}&0.406723818145&{}:{}&-0.871323839029&. \end{alignedat} \]
2c (130)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{2c}}&{}\approx{}&1.438420640281&{}:{}&0.806422458090&{}:{}&-1.244843098371&,\\ A^\prime_{\mathbf{2c}}&{}\approx{}&1.375426647384&{}:{}&0.442845925067&{}:{}&-0.818272572451&,\\B^\prime_{\mathbf{2c}}&{}\approx{}&67.781251278963&{}:{}&58.462170210314&{}:{}&-125.243421489277&,\\C^\prime_{\mathbf{2c}}&{}\approx{}&0.783227248675&{}:{}&0.783227248675&{}:{}&-0.566454497351&,\\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.548560540122&{}:{}&1.442106609715&{}:{}&-1.990667149838&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.161529878283&{}:{}&0.565788409543&{}:{}&-0.727318287826&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.806470076964&{}:{}&0.819452174001&{}:{}&-1.625922250965&, \end{alignedat} \]
2c (130)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{2c}}&{}\approx{}&1.143057852177&{}:{}&-0.769865097862&{}:{}&0.626807245685&,\\ A^*_{\mathbf{2c}}&{}\approx{}&0.000000000000&{}:{}&5.381494871788&{}:{}&-4.381494871788&,\\B^*_{\mathbf{2c}}&{}\approx{}&0.645844620338&{}:{}&0.000000000000&{}:{}&0.354155379662&,\\C^*_{\mathbf{2c}}&{}\approx{}&3.062915447742&{}:{}&-2.062915447742&{}:{}&0.000000000000&. \end{alignedat} \]
2c (130)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{2c}}&{}\approx{}&1.672613439834&{}:{}&1.126528029260&{}:{}&-1.799141469094&,\\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.548560540122&{}:{}&1.442106609715&{}:{}&-1.990667149838&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.161529878283&{}:{}&0.565788409543&{}:{}&-0.727318287826&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.806470076964&{}:{}&0.819452174001&{}:{}&-1.625922250965&,\\ A^*_{\mathbf{2c}}&{}\approx{}&0.000000000000&{}:{}&5.381494871788&{}:{}&-4.381494871788&,\\B^*_{\mathbf{2c}}&{}\approx{}&0.645844620338&{}:{}&0.000000000000&{}:{}&0.354155379662&,\\C^*_{\mathbf{2c}}&{}\approx{}&3.062915447742&{}:{}&-2.062915447742&{}:{}&0.000000000000&. \end{alignedat} \]
2c (130)