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 > 5c (312)

5c(312)

Malfatti circles

5c (312)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&6.568535592272&{}:{}&6.568535592272&{}:{}&-12.137071184544&, \\ P_{\mathbf{5c}}&{}\approx{}&-0.592493160293&{}:{}&-0.268767146754&{}:{}&1.861260307047&, \\ P^-_{\mathbf{5c}}&{}\approx{}&-0.256649091659&{}:{}&0.051894541629&{}:{}&1.204754550030&, \\ P^+_{\mathbf{5c}}&{}\approx{}&-0.963099231763&{}:{}&-0.622619364797&{}:{}&2.585718596560&, \\ Q_{\mathbf{5c}}&{}\approx{}&-1.108959378294&{}:{}&0.746899309312&{}:{}&1.362060068982&, \\ I^\prime_{\mathbf{5c}}&{}\approx{}&-1.359155018361&{}:{}&-0.915409495002&{}:{}&3.274564513363&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{5c}}\)
\(P^-_{\mathbf{5c}}\)
\(P^+_{\mathbf{5c}}\)
\(Q_{\mathbf{5c}}\)
\(I^\prime_{\mathbf{5c}}\)
5c (312)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5c}}&{}\approx{}&-56.920974110168&{}:{}&-68.322447379109&{}:{}&126.243421489277&,\\B^\prime_{\mathbf{5c}}&{}\approx{}&-0.984042025213&{}:{}&0.165769452762&{}:{}&1.818272572451&,\\C^\prime_{\mathbf{5c}}&{}\approx{}&-0.523335716201&{}:{}&-0.523335716201&{}:{}&2.046671432402&, \\ A^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.683285050440&{}:{}&-0.284090212417&{}:{}&1.967375262857&,\\B^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.952046047524&{}:{}&-1.038715091156&{}:{}&2.990761138679&,\\C^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-1.778257580008&{}:{}&-0.806654402114&{}:{}&3.584911982122&, \\ A^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.324751554676&{}:{}&0.054706898814&{}:{}&1.270044655861&,\\B^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.400983729597&{}:{}&-0.481302194728&{}:{}&1.882285924325&,\\C^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.704189577613&{}:{}&0.142387394064&{}:{}&1.561802183549&, \\ A^*_{\mathbf{5c}}&{}\approx{}&0.000000000000&{}:{}&0.354155379662&{}:{}&0.645844620338&,\\B^*_{\mathbf{5c}}&{}\approx{}&-4.381494871788&{}:{}&0.000000000000&{}:{}&5.381494871788&,\\C^*_{\mathbf{5c}}&{}\approx{}&3.062915447742&{}:{}&-2.062915447742&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5c}}}{B^\prime_{\mathbf{5c}}}{C^\prime_{\mathbf{5c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5c}}}{B^{\prime\prime}_{\mathbf{5c}}}{C^{\prime\prime}_{\mathbf{5c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5c}}}{B^{\prime\prime\prime}_{\mathbf{5c}}}{C^{\prime\prime\prime}_{\mathbf{5c}}}\)
\(\triangle{A^*_{\mathbf{5c}}}{B^*_{\mathbf{5c}}}{C^*_{\mathbf{5c}}}\)
5c (312)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5c}}}}&{}\approx{}&-10.401473268942&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5c}}}}&{}\approx{}&-0.149811477976&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5c}}}}&{}\approx{}&-0.079673118742&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&6.568535592272&{}:{}&6.568535592272&{}:{}&-12.137071184544&,\\ A^\prime_{\mathbf{5c}}&{}\approx{}&-56.920974110168&{}:{}&-68.322447379109&{}:{}&126.243421489277&,\\B^\prime_{\mathbf{5c}}&{}\approx{}&-0.984042025213&{}:{}&0.165769452762&{}:{}&1.818272572451&,\\C^\prime_{\mathbf{5c}}&{}\approx{}&-0.523335716201&{}:{}&-0.523335716201&{}:{}&2.046671432402&. \end{alignedat} \]
5c (312)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5c}}&{}\approx{}&-0.592493160293&{}:{}&-0.268767146754&{}:{}&1.861260307047&,\\ A^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.683285050440&{}:{}&-0.284090212417&{}:{}&1.967375262857&,\\B^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.952046047524&{}:{}&-1.038715091156&{}:{}&2.990761138679&,\\C^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-1.778257580008&{}:{}&-0.806654402114&{}:{}&3.584911982122&. \end{alignedat} \]
5c (312)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5c}}&{}\approx{}&-0.256649091659&{}:{}&0.051894541629&{}:{}&1.204754550030&,\\ A^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.324751554676&{}:{}&0.054706898814&{}:{}&1.270044655861&,\\B^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.400983729597&{}:{}&-0.481302194728&{}:{}&1.882285924325&,\\C^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.704189577613&{}:{}&0.142387394064&{}:{}&1.561802183549&. \end{alignedat} \]
5c (312)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5c}}&{}\approx{}&-0.963099231763&{}:{}&-0.622619364797&{}:{}&2.585718596560&,\\ A^\prime_{\mathbf{5c}}&{}\approx{}&-56.920974110168&{}:{}&-68.322447379109&{}:{}&126.243421489277&,\\B^\prime_{\mathbf{5c}}&{}\approx{}&-0.984042025213&{}:{}&0.165769452762&{}:{}&1.818272572451&,\\C^\prime_{\mathbf{5c}}&{}\approx{}&-0.523335716201&{}:{}&-0.523335716201&{}:{}&2.046671432402&,\\ A^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.683285050440&{}:{}&-0.284090212417&{}:{}&1.967375262857&,\\B^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.952046047524&{}:{}&-1.038715091156&{}:{}&2.990761138679&,\\C^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-1.778257580008&{}:{}&-0.806654402114&{}:{}&3.584911982122&, \end{alignedat} \]
5c (312)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5c}}&{}\approx{}&-1.108959378294&{}:{}&0.746899309312&{}:{}&1.362060068982&,\\ A^*_{\mathbf{5c}}&{}\approx{}&0.000000000000&{}:{}&0.354155379662&{}:{}&0.645844620338&,\\B^*_{\mathbf{5c}}&{}\approx{}&-4.381494871788&{}:{}&0.000000000000&{}:{}&5.381494871788&,\\C^*_{\mathbf{5c}}&{}\approx{}&3.062915447742&{}:{}&-2.062915447742&{}:{}&0.000000000000&. \end{alignedat} \]
5c (312)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5c}}&{}\approx{}&-1.359155018361&{}:{}&-0.915409495002&{}:{}&3.274564513363&,\\ A^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.683285050440&{}:{}&-0.284090212417&{}:{}&1.967375262857&,\\B^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.952046047524&{}:{}&-1.038715091156&{}:{}&2.990761138679&,\\C^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-1.778257580008&{}:{}&-0.806654402114&{}:{}&3.584911982122&,\\ A^*_{\mathbf{5c}}&{}\approx{}&0.000000000000&{}:{}&0.354155379662&{}:{}&0.645844620338&,\\B^*_{\mathbf{5c}}&{}\approx{}&-4.381494871788&{}:{}&0.000000000000&{}:{}&5.381494871788&,\\C^*_{\mathbf{5c}}&{}\approx{}&3.062915447742&{}:{}&-2.062915447742&{}:{}&0.000000000000&. \end{alignedat} \]
5c (312)