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 > 4c (310)

4c(310)

Malfatti circles

4c (310)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&6.568535592272&{}:{}&6.568535592272&{}:{}&-12.137071184544&, \\ P_{\mathbf{4c}}&{}\approx{}&0.548216543556&{}:{}&1.208535181249&{}:{}&-0.756751724805&, \\ P^-_{\mathbf{4c}}&{}\approx{}&0.265784096255&{}:{}&0.957080395984&{}:{}&-0.222864492240&, \\ P^+_{\mathbf{4c}}&{}\approx{}&0.806422458090&{}:{}&1.438420640281&{}:{}&-1.244843098371&, \\ Q_{\mathbf{4c}}&{}\approx{}&-0.769865097862&{}:{}&1.143057852177&{}:{}&0.626807245685&, \\ I^\prime_{\mathbf{4c}}&{}\approx{}&1.126528029260&{}:{}&1.672613439834&{}:{}&-1.799141469094&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{4c}}\)
\(P^-_{\mathbf{4c}}\)
\(P^+_{\mathbf{4c}}\)
\(Q_{\mathbf{4c}}\)
\(I^\prime_{\mathbf{4c}}\)
4c (310)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4c}}&{}\approx{}&58.462170210314&{}:{}&67.781251278963&{}:{}&-125.243421489277&,\\B^\prime_{\mathbf{4c}}&{}\approx{}&0.442845925067&{}:{}&1.375426647384&{}:{}&-0.818272572451&,\\C^\prime_{\mathbf{4c}}&{}\approx{}&0.783227248675&{}:{}&0.783227248675&{}:{}&-0.566454497351&, \\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.565788409543&{}:{}&1.161529878283&{}:{}&-0.727318287826&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&1.442106609715&{}:{}&1.548560540122&{}:{}&-1.990667149838&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.819452174001&{}:{}&1.806470076964&{}:{}&-1.625922250965&, \\ A^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.295626259650&{}:{}&0.918179918055&{}:{}&-0.213806177705&,\\B^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.757067273970&{}:{}&0.877746531603&{}:{}&-0.634813805573&,\\C^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.406723818145&{}:{}&1.464600020884&{}:{}&-0.871323839029&, \\ A^*_{\mathbf{4c}}&{}\approx{}&0.000000000000&{}:{}&0.645844620338&{}:{}&0.354155379662&,\\B^*_{\mathbf{4c}}&{}\approx{}&5.381494871788&{}:{}&0.000000000000&{}:{}&-4.381494871788&,\\C^*_{\mathbf{4c}}&{}\approx{}&-2.062915447742&{}:{}&3.062915447742&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4c}}}{B^\prime_{\mathbf{4c}}}{C^\prime_{\mathbf{4c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4c}}}{B^{\prime\prime}_{\mathbf{4c}}}{C^{\prime\prime}_{\mathbf{4c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4c}}}{B^{\prime\prime\prime}_{\mathbf{4c}}}{C^{\prime\prime\prime}_{\mathbf{4c}}}\)
\(\triangle{A^*_{\mathbf{4c}}}{B^*_{\mathbf{4c}}}{C^*_{\mathbf{4c}}}\)
4c (310)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4c}}}}&{}\approx{}&10.319081068649&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4c}}}}&{}\approx{}&0.067419277683&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4c}}}}&{}\approx{}&0.119239248638&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&6.568535592272&{}:{}&6.568535592272&{}:{}&-12.137071184544&,\\ A^\prime_{\mathbf{4c}}&{}\approx{}&58.462170210314&{}:{}&67.781251278963&{}:{}&-125.243421489277&,\\B^\prime_{\mathbf{4c}}&{}\approx{}&0.442845925067&{}:{}&1.375426647384&{}:{}&-0.818272572451&,\\C^\prime_{\mathbf{4c}}&{}\approx{}&0.783227248675&{}:{}&0.783227248675&{}:{}&-0.566454497351&. \end{alignedat} \]
4c (310)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4c}}&{}\approx{}&0.548216543556&{}:{}&1.208535181249&{}:{}&-0.756751724805&,\\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.565788409543&{}:{}&1.161529878283&{}:{}&-0.727318287826&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&1.442106609715&{}:{}&1.548560540122&{}:{}&-1.990667149838&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.819452174001&{}:{}&1.806470076964&{}:{}&-1.625922250965&. \end{alignedat} \]
4c (310)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4c}}&{}\approx{}&0.265784096255&{}:{}&0.957080395984&{}:{}&-0.222864492240&,\\ A^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.295626259650&{}:{}&0.918179918055&{}:{}&-0.213806177705&,\\B^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.757067273970&{}:{}&0.877746531603&{}:{}&-0.634813805573&,\\C^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.406723818145&{}:{}&1.464600020884&{}:{}&-0.871323839029&. \end{alignedat} \]
4c (310)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4c}}&{}\approx{}&0.806422458090&{}:{}&1.438420640281&{}:{}&-1.244843098371&,\\ A^\prime_{\mathbf{4c}}&{}\approx{}&58.462170210314&{}:{}&67.781251278963&{}:{}&-125.243421489277&,\\B^\prime_{\mathbf{4c}}&{}\approx{}&0.442845925067&{}:{}&1.375426647384&{}:{}&-0.818272572451&,\\C^\prime_{\mathbf{4c}}&{}\approx{}&0.783227248675&{}:{}&0.783227248675&{}:{}&-0.566454497351&,\\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.565788409543&{}:{}&1.161529878283&{}:{}&-0.727318287826&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&1.442106609715&{}:{}&1.548560540122&{}:{}&-1.990667149838&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.819452174001&{}:{}&1.806470076964&{}:{}&-1.625922250965&, \end{alignedat} \]
4c (310)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4c}}&{}\approx{}&-0.769865097862&{}:{}&1.143057852177&{}:{}&0.626807245685&,\\ A^*_{\mathbf{4c}}&{}\approx{}&0.000000000000&{}:{}&0.645844620338&{}:{}&0.354155379662&,\\B^*_{\mathbf{4c}}&{}\approx{}&5.381494871788&{}:{}&0.000000000000&{}:{}&-4.381494871788&,\\C^*_{\mathbf{4c}}&{}\approx{}&-2.062915447742&{}:{}&3.062915447742&{}:{}&0.000000000000&. \end{alignedat} \]
4c (310)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4c}}&{}\approx{}&1.126528029260&{}:{}&1.672613439834&{}:{}&-1.799141469094&,\\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.565788409543&{}:{}&1.161529878283&{}:{}&-0.727318287826&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&1.442106609715&{}:{}&1.548560540122&{}:{}&-1.990667149838&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.819452174001&{}:{}&1.806470076964&{}:{}&-1.625922250965&,\\ A^*_{\mathbf{4c}}&{}\approx{}&0.000000000000&{}:{}&0.645844620338&{}:{}&0.354155379662&,\\B^*_{\mathbf{4c}}&{}\approx{}&5.381494871788&{}:{}&0.000000000000&{}:{}&-4.381494871788&,\\C^*_{\mathbf{4c}}&{}\approx{}&-2.062915447742&{}:{}&3.062915447742&{}:{}&0.000000000000&. \end{alignedat} \]
4c (310)