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 > 6c (330)

6c(330)

Malfatti circles

6c (330)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&6.568535592272&{}:{}&6.568535592272&{}:{}&-12.137071184544&, \\ P_{\mathbf{6c}}&{}\approx{}&0.727893522251&{}:{}&0.727893522251&{}:{}&-0.455787044502&, \\ P^-_{\mathbf{6c}}&{}\approx{}&0.484073871176&{}:{}&0.484073871176&{}:{}&0.031852257648&, \\ P^+_{\mathbf{6c}}&{}\approx{}&0.952925126855&{}:{}&0.952925126855&{}:{}&-0.905850253710&, \\ Q_{\mathbf{6c}}&{}\approx{}&0.392409230104&{}:{}&0.392409230104&{}:{}&0.215181539792&, \\ I^\prime_{\mathbf{6c}}&{}\approx{}&1.081837671059&{}:{}&1.081837671059&{}:{}&-1.163675342118&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{6c}}\)
\(P^-_{\mathbf{6c}}\)
\(P^+_{\mathbf{6c}}\)
\(Q_{\mathbf{6c}}\)
\(I^\prime_{\mathbf{6c}}\)
6c (330)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6c}}&{}\approx{}&1.923879532511&{}:{}&1.089790213552&{}:{}&-2.013669746063&,\\B^\prime_{\mathbf{6c}}&{}\approx{}&1.089790213552&{}:{}&1.923879532511&{}:{}&-2.013669746063&,\\C^\prime_{\mathbf{6c}}&{}\approx{}&0.472553645310&{}:{}&0.472553645310&{}:{}&0.054892709380&, \\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.659245831340&{}:{}&0.911528288850&{}:{}&-0.570774120189&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.911528288850&{}:{}&0.659245831340&{}:{}&-0.570774120189&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&1.506834873031&{}:{}&1.506834873031&{}:{}&-2.013669746063&, \\ A^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.347038012205&{}:{}&0.612649407548&{}:{}&0.040312580247&,\\B^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.612649407548&{}:{}&0.347038012205&{}:{}&0.040312580247&,\\C^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&1.048956292708&{}:{}&1.048956292708&{}:{}&-1.097912585416&, \\ A^*_{\mathbf{6c}}&{}\approx{}&0.000000000000&{}:{}&0.645844620338&{}:{}&0.354155379662&,\\B^*_{\mathbf{6c}}&{}\approx{}&0.645844620338&{}:{}&0.000000000000&{}:{}&0.354155379662&,\\C^*_{\mathbf{6c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6c}}}{B^\prime_{\mathbf{6c}}}{C^\prime_{\mathbf{6c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6c}}}{B^{\prime\prime}_{\mathbf{6c}}}{C^{\prime\prime}_{\mathbf{6c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6c}}}{B^{\prime\prime\prime}_{\mathbf{6c}}}{C^{\prime\prime\prime}_{\mathbf{6c}}}\)
\(\triangle{A^*_{\mathbf{6c}}}{B^*_{\mathbf{6c}}}{C^*_{\mathbf{6c}}}\)
6c (330)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6c}}}}&{}\approx{}&0.165910681040&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6c}}}}&{}\approx{}&0.165910681040&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6c}}}}&{}\approx{}&0.071942008789&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&6.568535592272&{}:{}&6.568535592272&{}:{}&-12.137071184544&,\\ A^\prime_{\mathbf{6c}}&{}\approx{}&1.923879532511&{}:{}&1.089790213552&{}:{}&-2.013669746063&,\\B^\prime_{\mathbf{6c}}&{}\approx{}&1.089790213552&{}:{}&1.923879532511&{}:{}&-2.013669746063&,\\C^\prime_{\mathbf{6c}}&{}\approx{}&0.472553645310&{}:{}&0.472553645310&{}:{}&0.054892709380&. \end{alignedat} \]
6c (330)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6c}}&{}\approx{}&0.727893522251&{}:{}&0.727893522251&{}:{}&-0.455787044502&,\\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.659245831340&{}:{}&0.911528288850&{}:{}&-0.570774120189&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.911528288850&{}:{}&0.659245831340&{}:{}&-0.570774120189&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&1.506834873031&{}:{}&1.506834873031&{}:{}&-2.013669746063&. \end{alignedat} \]
6c (330)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6c}}&{}\approx{}&0.484073871176&{}:{}&0.484073871176&{}:{}&0.031852257648&,\\ A^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.347038012205&{}:{}&0.612649407548&{}:{}&0.040312580247&,\\B^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.612649407548&{}:{}&0.347038012205&{}:{}&0.040312580247&,\\C^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&1.048956292708&{}:{}&1.048956292708&{}:{}&-1.097912585416&. \end{alignedat} \]
6c (330)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6c}}&{}\approx{}&0.952925126855&{}:{}&0.952925126855&{}:{}&-0.905850253710&,\\ A^\prime_{\mathbf{6c}}&{}\approx{}&1.923879532511&{}:{}&1.089790213552&{}:{}&-2.013669746063&,\\B^\prime_{\mathbf{6c}}&{}\approx{}&1.089790213552&{}:{}&1.923879532511&{}:{}&-2.013669746063&,\\C^\prime_{\mathbf{6c}}&{}\approx{}&0.472553645310&{}:{}&0.472553645310&{}:{}&0.054892709380&,\\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.659245831340&{}:{}&0.911528288850&{}:{}&-0.570774120189&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.911528288850&{}:{}&0.659245831340&{}:{}&-0.570774120189&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&1.506834873031&{}:{}&1.506834873031&{}:{}&-2.013669746063&, \end{alignedat} \]
6c (330)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6c}}&{}\approx{}&0.392409230104&{}:{}&0.392409230104&{}:{}&0.215181539792&,\\ A^*_{\mathbf{6c}}&{}\approx{}&0.000000000000&{}:{}&0.645844620338&{}:{}&0.354155379662&,\\B^*_{\mathbf{6c}}&{}\approx{}&0.645844620338&{}:{}&0.000000000000&{}:{}&0.354155379662&,\\C^*_{\mathbf{6c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
6c (330)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6c}}&{}\approx{}&1.081837671059&{}:{}&1.081837671059&{}:{}&-1.163675342118&,\\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.659245831340&{}:{}&0.911528288850&{}:{}&-0.570774120189&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.911528288850&{}:{}&0.659245831340&{}:{}&-0.570774120189&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&1.506834873031&{}:{}&1.506834873031&{}:{}&-2.013669746063&,\\ A^*_{\mathbf{6c}}&{}\approx{}&0.000000000000&{}:{}&0.645844620338&{}:{}&0.354155379662&,\\B^*_{\mathbf{6c}}&{}\approx{}&0.645844620338&{}:{}&0.000000000000&{}:{}&0.354155379662&,\\C^*_{\mathbf{6c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
6c (330)