Derousseau's Generalization of the Malfatti circles

Pythagorean Triangle U

\(C=90\degree\).   \(a:b:c=5:12:13\).


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5b(303)

Malfatti circles

5b (303)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.833333333333&{}:{}&-2.000000000000&{}:{}&2.166666666667&, \\ P_{\mathbf{5b}}&{}\approx{}&0.049692614584&{}:{}&-0.022962275815&{}:{}&0.973269661231&, \\ P^-_{\mathbf{5b}}&{}\approx{}&0.013052902521&{}:{}&0.069475612134&{}:{}&0.917471485345&, \\ P^+_{\mathbf{5b}}&{}\approx{}&0.083199086026&{}:{}&-0.107495348767&{}:{}&1.024296262741&, \\ Q_{\mathbf{5b}}&{}\approx{}&0.309847094033&{}:{}&0.076909207526&{}:{}&0.613243698441&, \\ I^\prime_{\mathbf{5b}}&{}\approx{}&0.141184169980&{}:{}&-0.148680210238&{}:{}&1.007496040259&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{5b}}\)
\(P^-_{\mathbf{5b}}\)
\(P^+_{\mathbf{5b}}\)
\(Q_{\mathbf{5b}}\)
\(I^\prime_{\mathbf{5b}}\)
5b (303)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5b}}&{}\approx{}&0.626172792756&{}:{}&-4.485926486933&{}:{}&4.859753694177&,\\B^\prime_{\mathbf{5b}}&{}\approx{}&0.185766159067&{}:{}&0.331241827360&{}:{}&0.482992013574&,\\C^\prime_{\mathbf{5b}}&{}\approx{}&0.045152490866&{}:{}&-0.108365978077&{}:{}&1.063213487212&, \\ A^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.072647270395&{}:{}&-0.022407622504&{}:{}&0.949760352109&,\\B^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.058857067528&{}:{}&-0.211619880555&{}:{}&1.152762813027&,\\C^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.225579561844&{}:{}&-0.104237222386&{}:{}&0.878657660542&, \\ A^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.037979081144&{}:{}&0.067720947145&{}:{}&0.894299971712&,\\B^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.015594625615&{}:{}&-0.111720510552&{}:{}&1.096125884937&,\\C^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.071004268062&{}:{}&0.377928585606&{}:{}&0.551067146332&, \\ A^*_{\mathbf{5b}}&{}\approx{}&0.000000000000&{}:{}&0.111437924641&{}:{}&0.888562075359&,\\B^*_{\mathbf{5b}}&{}\approx{}&0.335662641811&{}:{}&0.000000000000&{}:{}&0.664337358189&,\\C^*_{\mathbf{5b}}&{}\approx{}&0.801142975005&{}:{}&0.198857024995&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5b}}}{B^\prime_{\mathbf{5b}}}{C^\prime_{\mathbf{5b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5b}}}{B^{\prime\prime}_{\mathbf{5b}}}{C^{\prime\prime}_{\mathbf{5b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5b}}}{B^{\prime\prime\prime}_{\mathbf{5b}}}{C^{\prime\prime\prime}_{\mathbf{5b}}}\)
\(\triangle{A^*_{\mathbf{5b}}}{B^*_{\mathbf{5b}}}{C^*_{\mathbf{5b}}}\)
5b (303)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5b}}}}&{}\approx{}&2.242963243466&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5b}}}}&{}\approx{}&0.222919390880&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5b}}}}&{}\approx{}&0.054182989039&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.833333333333&{}:{}&-2.000000000000&{}:{}&2.166666666667&,\\ A^\prime_{\mathbf{5b}}&{}\approx{}&0.626172792756&{}:{}&-4.485926486933&{}:{}&4.859753694177&,\\B^\prime_{\mathbf{5b}}&{}\approx{}&0.185766159067&{}:{}&0.331241827360&{}:{}&0.482992013574&,\\C^\prime_{\mathbf{5b}}&{}\approx{}&0.045152490866&{}:{}&-0.108365978077&{}:{}&1.063213487212&. \end{alignedat} \]
5b (303)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5b}}&{}\approx{}&0.049692614584&{}:{}&-0.022962275815&{}:{}&0.973269661231&,\\ A^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.072647270395&{}:{}&-0.022407622504&{}:{}&0.949760352109&,\\B^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.058857067528&{}:{}&-0.211619880555&{}:{}&1.152762813027&,\\C^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.225579561844&{}:{}&-0.104237222386&{}:{}&0.878657660542&. \end{alignedat} \]
5b (303)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5b}}&{}\approx{}&0.013052902521&{}:{}&0.069475612134&{}:{}&0.917471485345&,\\ A^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.037979081144&{}:{}&0.067720947145&{}:{}&0.894299971712&,\\B^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.015594625615&{}:{}&-0.111720510552&{}:{}&1.096125884937&,\\C^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.071004268062&{}:{}&0.377928585606&{}:{}&0.551067146332&. \end{alignedat} \]
5b (303)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5b}}&{}\approx{}&0.083199086026&{}:{}&-0.107495348767&{}:{}&1.024296262741&,\\ A^\prime_{\mathbf{5b}}&{}\approx{}&0.626172792756&{}:{}&-4.485926486933&{}:{}&4.859753694177&,\\B^\prime_{\mathbf{5b}}&{}\approx{}&0.185766159067&{}:{}&0.331241827360&{}:{}&0.482992013574&,\\C^\prime_{\mathbf{5b}}&{}\approx{}&0.045152490866&{}:{}&-0.108365978077&{}:{}&1.063213487212&,\\ A^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.072647270395&{}:{}&-0.022407622504&{}:{}&0.949760352109&,\\B^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.058857067528&{}:{}&-0.211619880555&{}:{}&1.152762813027&,\\C^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.225579561844&{}:{}&-0.104237222386&{}:{}&0.878657660542&, \end{alignedat} \]
5b (303)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5b}}&{}\approx{}&0.309847094033&{}:{}&0.076909207526&{}:{}&0.613243698441&,\\ A^*_{\mathbf{5b}}&{}\approx{}&0.000000000000&{}:{}&0.111437924641&{}:{}&0.888562075359&,\\B^*_{\mathbf{5b}}&{}\approx{}&0.335662641811&{}:{}&0.000000000000&{}:{}&0.664337358189&,\\C^*_{\mathbf{5b}}&{}\approx{}&0.801142975005&{}:{}&0.198857024995&{}:{}&0.000000000000&. \end{alignedat} \]
5b (303)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5b}}&{}\approx{}&0.141184169980&{}:{}&-0.148680210238&{}:{}&1.007496040259&,\\ A^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.072647270395&{}:{}&-0.022407622504&{}:{}&0.949760352109&,\\B^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.058857067528&{}:{}&-0.211619880555&{}:{}&1.152762813027&,\\C^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.225579561844&{}:{}&-0.104237222386&{}:{}&0.878657660542&,\\ A^*_{\mathbf{5b}}&{}\approx{}&0.000000000000&{}:{}&0.111437924641&{}:{}&0.888562075359&,\\B^*_{\mathbf{5b}}&{}\approx{}&0.335662641811&{}:{}&0.000000000000&{}:{}&0.664337358189&,\\C^*_{\mathbf{5b}}&{}\approx{}&0.801142975005&{}:{}&0.198857024995&{}:{}&0.000000000000&. \end{alignedat} \]
5b (303)