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 > 1c (112)

1c(112)

Malfatti circles

1c (112)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&6.568535592272&{}:{}&6.568535592272&{}:{}&-12.137071184544&, \\ P_{\mathbf{1c}}&{}\approx{}&-0.876114078250&{}:{}&-0.876114078250&{}:{}&2.752228156499&, \\ P^-_{\mathbf{1c}}&{}\approx{}&-0.150265610510&{}:{}&-0.150265610510&{}:{}&1.300531221021&, \\ P^+_{\mathbf{1c}}&{}\approx{}&-1.777787726467&{}:{}&-1.777787726467&{}:{}&4.555575452935&, \\ Q_{\mathbf{1c}}&{}\approx{}&1.295727196969&{}:{}&1.295727196969&{}:{}&-1.591454393939&, \\ I^\prime_{\mathbf{1c}}&{}\approx{}&-2.443405087842&{}:{}&-2.443405087842&{}:{}&5.886810175683&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{1c}}\)
\(P^-_{\mathbf{1c}}\)
\(P^+_{\mathbf{1c}}\)
\(Q_{\mathbf{1c}}\)
\(I^\prime_{\mathbf{1c}}\)
1c (112)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1c}}&{}\approx{}&-0.382683432365&{}:{}&-1.630986313698&{}:{}&3.013669746063&,\\B^\prime_{\mathbf{1c}}&{}\approx{}&-1.630986313698&{}:{}&-0.382683432365&{}:{}&3.013669746063&,\\C^\prime_{\mathbf{1c}}&{}\approx{}&-32.549704743153&{}:{}&-32.549704743153&{}:{}&66.099409486307&, \\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-3.106322389017&{}:{}&-1.917584275156&{}:{}&6.023906664173&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.917584275156&{}:{}&-3.106322389017&{}:{}&6.023906664173&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.006834873031&{}:{}&-1.006834873031&{}:{}&3.013669746063&, \\ A^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.256140786156&{}:{}&-0.294732250996&{}:{}&2.550873037152&,\\B^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.294732250996&{}:{}&-1.256140786156&{}:{}&2.550873037152&,\\C^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.170209910570&{}:{}&-0.170209910570&{}:{}&1.340419821139&, \\ A^*_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&-4.381494871788&{}:{}&5.381494871788&,\\B^*_{\mathbf{1c}}&{}\approx{}&-4.381494871788&{}:{}&0.000000000000&{}:{}&5.381494871788&,\\C^*_{\mathbf{1c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1c}}}{B^\prime_{\mathbf{1c}}}{C^\prime_{\mathbf{1c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1c}}}{B^{\prime\prime}_{\mathbf{1c}}}{C^{\prime\prime}_{\mathbf{1c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1c}}}{B^{\prime\prime\prime}_{\mathbf{1c}}}{C^{\prime\prime\prime}_{\mathbf{1c}}}\)
\(\triangle{A^*_{\mathbf{1c}}}{B^*_{\mathbf{1c}}}{C^*_{\mathbf{1c}}}\)
1c (112)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1c}}}}&{}\approx{}&-0.248302881333&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1c}}}}&{}\approx{}&-0.248302881333&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1c}}}}&{}\approx{}&-4.955397483337&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&6.568535592272&{}:{}&6.568535592272&{}:{}&-12.137071184544&,\\ A^\prime_{\mathbf{1c}}&{}\approx{}&-0.382683432365&{}:{}&-1.630986313698&{}:{}&3.013669746063&,\\B^\prime_{\mathbf{1c}}&{}\approx{}&-1.630986313698&{}:{}&-0.382683432365&{}:{}&3.013669746063&,\\C^\prime_{\mathbf{1c}}&{}\approx{}&-32.549704743153&{}:{}&-32.549704743153&{}:{}&66.099409486307&. \end{alignedat} \]
1c (112)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1c}}&{}\approx{}&-0.876114078250&{}:{}&-0.876114078250&{}:{}&2.752228156499&,\\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-3.106322389017&{}:{}&-1.917584275156&{}:{}&6.023906664173&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.917584275156&{}:{}&-3.106322389017&{}:{}&6.023906664173&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.006834873031&{}:{}&-1.006834873031&{}:{}&3.013669746063&. \end{alignedat} \]
1c (112)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1c}}&{}\approx{}&-0.150265610510&{}:{}&-0.150265610510&{}:{}&1.300531221021&,\\ A^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.256140786156&{}:{}&-0.294732250996&{}:{}&2.550873037152&,\\B^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.294732250996&{}:{}&-1.256140786156&{}:{}&2.550873037152&,\\C^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.170209910570&{}:{}&-0.170209910570&{}:{}&1.340419821139&. \end{alignedat} \]
1c (112)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1c}}&{}\approx{}&-1.777787726467&{}:{}&-1.777787726467&{}:{}&4.555575452935&,\\ A^\prime_{\mathbf{1c}}&{}\approx{}&-0.382683432365&{}:{}&-1.630986313698&{}:{}&3.013669746063&,\\B^\prime_{\mathbf{1c}}&{}\approx{}&-1.630986313698&{}:{}&-0.382683432365&{}:{}&3.013669746063&,\\C^\prime_{\mathbf{1c}}&{}\approx{}&-32.549704743153&{}:{}&-32.549704743153&{}:{}&66.099409486307&,\\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-3.106322389017&{}:{}&-1.917584275156&{}:{}&6.023906664173&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.917584275156&{}:{}&-3.106322389017&{}:{}&6.023906664173&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.006834873031&{}:{}&-1.006834873031&{}:{}&3.013669746063&, \end{alignedat} \]
1c (112)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1c}}&{}\approx{}&1.295727196969&{}:{}&1.295727196969&{}:{}&-1.591454393939&,\\ A^*_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&-4.381494871788&{}:{}&5.381494871788&,\\B^*_{\mathbf{1c}}&{}\approx{}&-4.381494871788&{}:{}&0.000000000000&{}:{}&5.381494871788&,\\C^*_{\mathbf{1c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
1c (112)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1c}}&{}\approx{}&-2.443405087842&{}:{}&-2.443405087842&{}:{}&5.886810175683&,\\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-3.106322389017&{}:{}&-1.917584275156&{}:{}&6.023906664173&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.917584275156&{}:{}&-3.106322389017&{}:{}&6.023906664173&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.006834873031&{}:{}&-1.006834873031&{}:{}&3.013669746063&,\\ A^*_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&-4.381494871788&{}:{}&5.381494871788&,\\B^*_{\mathbf{1c}}&{}\approx{}&-4.381494871788&{}:{}&0.000000000000&{}:{}&5.381494871788&,\\C^*_{\mathbf{1c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
1c (112)