Derousseau's Generalization of the Malfatti circles

Pythagorean Triangle U

\(C=90\degree\).   \(a:b:c=5:12:13\).


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6b(321)

Malfatti circles

6b (321)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.833333333333&{}:{}&-2.000000000000&{}:{}&2.166666666667&, \\ P_{\mathbf{6b}}&{}\approx{}&-0.018723845326&{}:{}&1.029519118620&{}:{}&-0.010795273294&, \\ P^-_{\mathbf{6b}}&{}\approx{}&0.007295688553&{}:{}&0.937005755293&{}:{}&0.055698556154&, \\ P^+_{\mathbf{6b}}&{}\approx{}&-0.046435881247&{}:{}&1.128050232191&{}:{}&-0.081614350944&, \\ Q_{\mathbf{6b}}&{}\approx{}&-0.488471521915&{}:{}&1.322599344626&{}:{}&0.165872177290&, \\ I^\prime_{\mathbf{6b}}&{}\approx{}&-0.107954717346&{}:{}&1.240129193221&{}:{}&-0.132174475875&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{6b}}\)
\(P^-_{\mathbf{6b}}\)
\(P^+_{\mathbf{6b}}\)
\(Q_{\mathbf{6b}}\)
\(I^\prime_{\mathbf{6b}}\)
6b (321)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6b}}&{}\approx{}&3.306997126177&{}:{}&27.683965514118&{}:{}&-29.990962640295&,\\B^\prime_{\mathbf{6b}}&{}\approx{}&-0.030101660577&{}:{}&1.108365978077&{}:{}&-0.078264317500&,\\C^\prime_{\mathbf{6b}}&{}\approx{}&-0.278649238600&{}:{}&0.668758172640&{}:{}&0.609891065960&, \\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.054333799984&{}:{}&1.065506426959&{}:{}&-0.011172626974&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.194074284281&{}:{}&1.305968230298&{}:{}&-0.111893946017&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.021415845031&{}:{}&1.177537066605&{}:{}&-0.156121221574&, \\ A^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.026309212305&{}:{}&0.968725155669&{}:{}&0.057584056636&,\\B^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.058803541204&{}:{}&0.492263871028&{}:{}&0.448932587768&,\\C^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.008308141310&{}:{}&1.067037904097&{}:{}&-0.075346045406&, \\ A^*_{\mathbf{6b}}&{}\approx{}&0.000000000000&{}:{}&0.888562075359&{}:{}&0.111437924641&,\\B^*_{\mathbf{6b}}&{}\approx{}&1.514173943788&{}:{}&0.000000000000&{}:{}&-0.514173943788&,\\C^*_{\mathbf{6b}}&{}\approx{}&-0.585607515558&{}:{}&1.585607515558&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6b}}}{B^\prime_{\mathbf{6b}}}{C^\prime_{\mathbf{6b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6b}}}{B^{\prime\prime}_{\mathbf{6b}}}{C^{\prime\prime}_{\mathbf{6b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6b}}}{B^{\prime\prime\prime}_{\mathbf{6b}}}{C^{\prime\prime\prime}_{\mathbf{6b}}}\)
\(\triangle{A^*_{\mathbf{6b}}}{B^*_{\mathbf{6b}}}{C^*_{\mathbf{6b}}}\)
6b (321)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6b}}}}&{}\approx{}&-13.841982757059&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6b}}}}&{}\approx{}&-0.036121992692&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6b}}}}&{}\approx{}&-0.334379086320&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.833333333333&{}:{}&-2.000000000000&{}:{}&2.166666666667&,\\ A^\prime_{\mathbf{6b}}&{}\approx{}&3.306997126177&{}:{}&27.683965514118&{}:{}&-29.990962640295&,\\B^\prime_{\mathbf{6b}}&{}\approx{}&-0.030101660577&{}:{}&1.108365978077&{}:{}&-0.078264317500&,\\C^\prime_{\mathbf{6b}}&{}\approx{}&-0.278649238600&{}:{}&0.668758172640&{}:{}&0.609891065960&. \end{alignedat} \]
6b (321)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6b}}&{}\approx{}&-0.018723845326&{}:{}&1.029519118620&{}:{}&-0.010795273294&,\\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.054333799984&{}:{}&1.065506426959&{}:{}&-0.011172626974&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.194074284281&{}:{}&1.305968230298&{}:{}&-0.111893946017&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.021415845031&{}:{}&1.177537066605&{}:{}&-0.156121221574&. \end{alignedat} \]
6b (321)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6b}}&{}\approx{}&0.007295688553&{}:{}&0.937005755293&{}:{}&0.055698556154&,\\ A^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.026309212305&{}:{}&0.968725155669&{}:{}&0.057584056636&,\\B^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.058803541204&{}:{}&0.492263871028&{}:{}&0.448932587768&,\\C^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.008308141310&{}:{}&1.067037904097&{}:{}&-0.075346045406&. \end{alignedat} \]
6b (321)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6b}}&{}\approx{}&-0.046435881247&{}:{}&1.128050232191&{}:{}&-0.081614350944&,\\ A^\prime_{\mathbf{6b}}&{}\approx{}&3.306997126177&{}:{}&27.683965514118&{}:{}&-29.990962640295&,\\B^\prime_{\mathbf{6b}}&{}\approx{}&-0.030101660577&{}:{}&1.108365978077&{}:{}&-0.078264317500&,\\C^\prime_{\mathbf{6b}}&{}\approx{}&-0.278649238600&{}:{}&0.668758172640&{}:{}&0.609891065960&,\\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.054333799984&{}:{}&1.065506426959&{}:{}&-0.011172626974&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.194074284281&{}:{}&1.305968230298&{}:{}&-0.111893946017&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.021415845031&{}:{}&1.177537066605&{}:{}&-0.156121221574&, \end{alignedat} \]
6b (321)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6b}}&{}\approx{}&-0.488471521915&{}:{}&1.322599344626&{}:{}&0.165872177290&,\\ A^*_{\mathbf{6b}}&{}\approx{}&0.000000000000&{}:{}&0.888562075359&{}:{}&0.111437924641&,\\B^*_{\mathbf{6b}}&{}\approx{}&1.514173943788&{}:{}&0.000000000000&{}:{}&-0.514173943788&,\\C^*_{\mathbf{6b}}&{}\approx{}&-0.585607515558&{}:{}&1.585607515558&{}:{}&0.000000000000&. \end{alignedat} \]
6b (321)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6b}}&{}\approx{}&-0.107954717346&{}:{}&1.240129193221&{}:{}&-0.132174475875&,\\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.054333799984&{}:{}&1.065506426959&{}:{}&-0.011172626974&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.194074284281&{}:{}&1.305968230298&{}:{}&-0.111893946017&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.021415845031&{}:{}&1.177537066605&{}:{}&-0.156121221574&,\\ A^*_{\mathbf{6b}}&{}\approx{}&0.000000000000&{}:{}&0.888562075359&{}:{}&0.111437924641&,\\B^*_{\mathbf{6b}}&{}\approx{}&1.514173943788&{}:{}&0.000000000000&{}:{}&-0.514173943788&,\\C^*_{\mathbf{6b}}&{}\approx{}&-0.585607515558&{}:{}&1.585607515558&{}:{}&0.000000000000&. \end{alignedat} \]
6b (321)