Derousseau's Generalization of the Malfatti circles

Pythagorean Triangle U

\(C=90\degree\).   \(a:b:c=5:12:13\).


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4b(301)

Malfatti circles

4b (301)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.833333333333&{}:{}&-2.000000000000&{}:{}&2.166666666667&, \\ P_{\mathbf{4b}}&{}\approx{}&0.023074334733&{}:{}&-0.023605453268&{}:{}&1.000531118535&, \\ P^-_{\mathbf{4b}}&{}\approx{}&-0.002727154777&{}:{}&0.039329883771&{}:{}&0.963397271006&, \\ P^+_{\mathbf{4b}}&{}\approx{}&0.047330990203&{}:{}&-0.082772610528&{}:{}&1.035441620325&, \\ Q_{\mathbf{4b}}&{}\approx{}&-0.432115884436&{}:{}&0.159592022007&{}:{}&1.272523862429&, \\ I^\prime_{\mathbf{4b}}&{}\approx{}&0.099493101869&{}:{}&-0.155897865706&{}:{}&1.056404763837&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{4b}}\)
\(P^-_{\mathbf{4b}}\)
\(P^+_{\mathbf{4b}}\)
\(Q_{\mathbf{4b}}\)
\(I^\prime_{\mathbf{4b}}\)
4b (301)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4b}}&{}\approx{}&-1.056997126177&{}:{}&-24.683965514118&{}:{}&26.740962640295&,\\B^\prime_{\mathbf{4b}}&{}\approx{}&0.196768327244&{}:{}&0.291634021923&{}:{}&0.511597650834&,\\C^\prime_{\mathbf{4b}}&{}\approx{}&0.028649238600&{}:{}&-0.068758172640&{}:{}&1.040108934040&, \\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.050016179840&{}:{}&-0.022954457508&{}:{}&0.972938277668&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.025633527638&{}:{}&-0.137134352061&{}:{}&1.111500824424&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.173232062988&{}:{}&-0.177219469797&{}:{}&1.003987406810&, \\ A^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.025781794028&{}:{}&0.038211679644&{}:{}&0.936006526328&,\\B^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&-0.003040363615&{}:{}&-0.071001357313&{}:{}&1.074041720929&,\\C^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&-0.025826063525&{}:{}&0.372452669442&{}:{}&0.653373394083&, \\ A^*_{\mathbf{4b}}&{}\approx{}&0.000000000000&{}:{}&0.111437924641&{}:{}&0.888562075359&,\\B^*_{\mathbf{4b}}&{}\approx{}&-0.514173943788&{}:{}&0.000000000000&{}:{}&1.514173943788&,\\C^*_{\mathbf{4b}}&{}\approx{}&1.585607515558&{}:{}&-0.585607515558&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4b}}}{B^\prime_{\mathbf{4b}}}{C^\prime_{\mathbf{4b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4b}}}{B^{\prime\prime}_{\mathbf{4b}}}{C^{\prime\prime}_{\mathbf{4b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4b}}}{B^{\prime\prime\prime}_{\mathbf{4b}}}{C^{\prime\prime\prime}_{\mathbf{4b}}}\)
\(\triangle{A^*_{\mathbf{4b}}}{B^*_{\mathbf{4b}}}{C^*_{\mathbf{4b}}}\)
4b (301)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4b}}}}&{}\approx{}&12.341982757059&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4b}}}}&{}\approx{}&0.236121992692&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4b}}}}&{}\approx{}&0.034379086320&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.833333333333&{}:{}&-2.000000000000&{}:{}&2.166666666667&,\\ A^\prime_{\mathbf{4b}}&{}\approx{}&-1.056997126177&{}:{}&-24.683965514118&{}:{}&26.740962640295&,\\B^\prime_{\mathbf{4b}}&{}\approx{}&0.196768327244&{}:{}&0.291634021923&{}:{}&0.511597650834&,\\C^\prime_{\mathbf{4b}}&{}\approx{}&0.028649238600&{}:{}&-0.068758172640&{}:{}&1.040108934040&. \end{alignedat} \]
4b (301)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4b}}&{}\approx{}&0.023074334733&{}:{}&-0.023605453268&{}:{}&1.000531118535&,\\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.050016179840&{}:{}&-0.022954457508&{}:{}&0.972938277668&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.025633527638&{}:{}&-0.137134352061&{}:{}&1.111500824424&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.173232062988&{}:{}&-0.177219469797&{}:{}&1.003987406810&. \end{alignedat} \]
4b (301)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4b}}&{}\approx{}&-0.002727154777&{}:{}&0.039329883771&{}:{}&0.963397271006&,\\ A^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.025781794028&{}:{}&0.038211679644&{}:{}&0.936006526328&,\\B^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&-0.003040363615&{}:{}&-0.071001357313&{}:{}&1.074041720929&,\\C^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&-0.025826063525&{}:{}&0.372452669442&{}:{}&0.653373394083&. \end{alignedat} \]
4b (301)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4b}}&{}\approx{}&0.047330990203&{}:{}&-0.082772610528&{}:{}&1.035441620325&,\\ A^\prime_{\mathbf{4b}}&{}\approx{}&-1.056997126177&{}:{}&-24.683965514118&{}:{}&26.740962640295&,\\B^\prime_{\mathbf{4b}}&{}\approx{}&0.196768327244&{}:{}&0.291634021923&{}:{}&0.511597650834&,\\C^\prime_{\mathbf{4b}}&{}\approx{}&0.028649238600&{}:{}&-0.068758172640&{}:{}&1.040108934040&,\\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.050016179840&{}:{}&-0.022954457508&{}:{}&0.972938277668&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.025633527638&{}:{}&-0.137134352061&{}:{}&1.111500824424&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.173232062988&{}:{}&-0.177219469797&{}:{}&1.003987406810&, \end{alignedat} \]
4b (301)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4b}}&{}\approx{}&-0.432115884436&{}:{}&0.159592022007&{}:{}&1.272523862429&,\\ A^*_{\mathbf{4b}}&{}\approx{}&0.000000000000&{}:{}&0.111437924641&{}:{}&0.888562075359&,\\B^*_{\mathbf{4b}}&{}\approx{}&-0.514173943788&{}:{}&0.000000000000&{}:{}&1.514173943788&,\\C^*_{\mathbf{4b}}&{}\approx{}&1.585607515558&{}:{}&-0.585607515558&{}:{}&0.000000000000&. \end{alignedat} \]
4b (301)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4b}}&{}\approx{}&0.099493101869&{}:{}&-0.155897865706&{}:{}&1.056404763837&,\\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.050016179840&{}:{}&-0.022954457508&{}:{}&0.972938277668&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.025633527638&{}:{}&-0.137134352061&{}:{}&1.111500824424&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.173232062988&{}:{}&-0.177219469797&{}:{}&1.003987406810&,\\ A^*_{\mathbf{4b}}&{}\approx{}&0.000000000000&{}:{}&0.111437924641&{}:{}&0.888562075359&,\\B^*_{\mathbf{4b}}&{}\approx{}&-0.514173943788&{}:{}&0.000000000000&{}:{}&1.514173943788&,\\C^*_{\mathbf{4b}}&{}\approx{}&1.585607515558&{}:{}&-0.585607515558&{}:{}&0.000000000000&. \end{alignedat} \]
4b (301)