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 > 3c (132)

3c(132)

Malfatti circles

3c (132)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&6.568535592272&{}:{}&6.568535592272&{}:{}&-12.137071184544&, \\ P_{\mathbf{3c}}&{}\approx{}&-0.268767146754&{}:{}&-0.592493160293&{}:{}&1.861260307047&, \\ P^-_{\mathbf{3c}}&{}\approx{}&0.051894541629&{}:{}&-0.256649091659&{}:{}&1.204754550030&, \\ P^+_{\mathbf{3c}}&{}\approx{}&-0.622619364797&{}:{}&-0.963099231763&{}:{}&2.585718596560&, \\ Q_{\mathbf{3c}}&{}\approx{}&0.746899309312&{}:{}&-1.108959378294&{}:{}&1.362060068982&, \\ I^\prime_{\mathbf{3c}}&{}\approx{}&-0.915409495002&{}:{}&-1.359155018361&{}:{}&3.274564513363&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{3c}}\)
\(P^-_{\mathbf{3c}}\)
\(P^+_{\mathbf{3c}}\)
\(Q_{\mathbf{3c}}\)
\(I^\prime_{\mathbf{3c}}\)
3c (132)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3c}}&{}\approx{}&0.165769452762&{}:{}&-0.984042025213&{}:{}&1.818272572451&,\\B^\prime_{\mathbf{3c}}&{}\approx{}&-68.322447379109&{}:{}&-56.920974110168&{}:{}&126.243421489277&,\\C^\prime_{\mathbf{3c}}&{}\approx{}&-0.523335716201&{}:{}&-0.523335716201&{}:{}&2.046671432402&, \\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-1.038715091156&{}:{}&-0.952046047524&{}:{}&2.990761138679&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.284090212417&{}:{}&-0.683285050440&{}:{}&1.967375262857&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.806654402114&{}:{}&-1.778257580008&{}:{}&3.584911982122&, \\ A^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.481302194728&{}:{}&-0.400983729597&{}:{}&1.882285924325&,\\B^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.054706898814&{}:{}&-0.324751554676&{}:{}&1.270044655861&,\\C^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.142387394064&{}:{}&-0.704189577613&{}:{}&1.561802183549&, \\ A^*_{\mathbf{3c}}&{}\approx{}&0.000000000000&{}:{}&-4.381494871788&{}:{}&5.381494871788&,\\B^*_{\mathbf{3c}}&{}\approx{}&0.354155379662&{}:{}&0.000000000000&{}:{}&0.645844620338&,\\C^*_{\mathbf{3c}}&{}\approx{}&-2.062915447742&{}:{}&3.062915447742&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3c}}}{B^\prime_{\mathbf{3c}}}{C^\prime_{\mathbf{3c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3c}}}{B^{\prime\prime}_{\mathbf{3c}}}{C^{\prime\prime}_{\mathbf{3c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3c}}}{B^{\prime\prime\prime}_{\mathbf{3c}}}{C^{\prime\prime\prime}_{\mathbf{3c}}}\)
\(\triangle{A^*_{\mathbf{3c}}}{B^*_{\mathbf{3c}}}{C^*_{\mathbf{3c}}}\)
3c (132)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3c}}}}&{}\approx{}&-0.149811477976&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3c}}}}&{}\approx{}&-10.401473268942&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3c}}}}&{}\approx{}&-0.079673118742&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&6.568535592272&{}:{}&6.568535592272&{}:{}&-12.137071184544&,\\ A^\prime_{\mathbf{3c}}&{}\approx{}&0.165769452762&{}:{}&-0.984042025213&{}:{}&1.818272572451&,\\B^\prime_{\mathbf{3c}}&{}\approx{}&-68.322447379109&{}:{}&-56.920974110168&{}:{}&126.243421489277&,\\C^\prime_{\mathbf{3c}}&{}\approx{}&-0.523335716201&{}:{}&-0.523335716201&{}:{}&2.046671432402&. \end{alignedat} \]
3c (132)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3c}}&{}\approx{}&-0.268767146754&{}:{}&-0.592493160293&{}:{}&1.861260307047&,\\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-1.038715091156&{}:{}&-0.952046047524&{}:{}&2.990761138679&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.284090212417&{}:{}&-0.683285050440&{}:{}&1.967375262857&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.806654402114&{}:{}&-1.778257580008&{}:{}&3.584911982122&. \end{alignedat} \]
3c (132)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3c}}&{}\approx{}&0.051894541629&{}:{}&-0.256649091659&{}:{}&1.204754550030&,\\ A^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.481302194728&{}:{}&-0.400983729597&{}:{}&1.882285924325&,\\B^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.054706898814&{}:{}&-0.324751554676&{}:{}&1.270044655861&,\\C^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.142387394064&{}:{}&-0.704189577613&{}:{}&1.561802183549&. \end{alignedat} \]
3c (132)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3c}}&{}\approx{}&-0.622619364797&{}:{}&-0.963099231763&{}:{}&2.585718596560&,\\ A^\prime_{\mathbf{3c}}&{}\approx{}&0.165769452762&{}:{}&-0.984042025213&{}:{}&1.818272572451&,\\B^\prime_{\mathbf{3c}}&{}\approx{}&-68.322447379109&{}:{}&-56.920974110168&{}:{}&126.243421489277&,\\C^\prime_{\mathbf{3c}}&{}\approx{}&-0.523335716201&{}:{}&-0.523335716201&{}:{}&2.046671432402&,\\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-1.038715091156&{}:{}&-0.952046047524&{}:{}&2.990761138679&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.284090212417&{}:{}&-0.683285050440&{}:{}&1.967375262857&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.806654402114&{}:{}&-1.778257580008&{}:{}&3.584911982122&, \end{alignedat} \]
3c (132)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3c}}&{}\approx{}&0.746899309312&{}:{}&-1.108959378294&{}:{}&1.362060068982&,\\ A^*_{\mathbf{3c}}&{}\approx{}&0.000000000000&{}:{}&-4.381494871788&{}:{}&5.381494871788&,\\B^*_{\mathbf{3c}}&{}\approx{}&0.354155379662&{}:{}&0.000000000000&{}:{}&0.645844620338&,\\C^*_{\mathbf{3c}}&{}\approx{}&-2.062915447742&{}:{}&3.062915447742&{}:{}&0.000000000000&. \end{alignedat} \]
3c (132)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3c}}&{}\approx{}&-0.915409495002&{}:{}&-1.359155018361&{}:{}&3.274564513363&,\\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-1.038715091156&{}:{}&-0.952046047524&{}:{}&2.990761138679&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.284090212417&{}:{}&-0.683285050440&{}:{}&1.967375262857&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.806654402114&{}:{}&-1.778257580008&{}:{}&3.584911982122&,\\ A^*_{\mathbf{3c}}&{}\approx{}&0.000000000000&{}:{}&-4.381494871788&{}:{}&5.381494871788&,\\B^*_{\mathbf{3c}}&{}\approx{}&0.354155379662&{}:{}&0.000000000000&{}:{}&0.645844620338&,\\C^*_{\mathbf{3c}}&{}\approx{}&-2.062915447742&{}:{}&3.062915447742&{}:{}&0.000000000000&. \end{alignedat} \]
3c (132)