Derousseau's Generalization of the Malfatti circles

Pythagorean Triangle U

\(C=90\degree\).   \(a:b:c=5:12:13\).


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7a(233)

Malfatti circles

7a (233)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.250000000000&{}:{}&0.600000000000&{}:{}&0.650000000000&, \\ P_{\mathbf{7a}}&{}\approx{}&1.000711398859&{}:{}&-0.000374709765&{}:{}&-0.000336689094&, \\ P^-_{\mathbf{7a}}&{}\approx{}&0.982660079590&{}:{}&0.008290403207&{}:{}&0.009049517202&, \\ P^+_{\mathbf{7a}}&{}\approx{}&1.019299269712&{}:{}&-0.009297381757&{}:{}&-0.010001887955&, \\ Q_{\mathbf{7a}}&{}\approx{}&0.914069378891&{}:{}&0.048439859439&{}:{}&0.037490761670&, \\ I^\prime_{\mathbf{7a}}&{}\approx{}&1.063325529863&{}:{}&-0.031876082646&{}:{}&-0.031449447216&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{7a}}\)
\(P^-_{\mathbf{7a}}\)
\(P^+_{\mathbf{7a}}\)
\(Q_{\mathbf{7a}}\)
\(I^\prime_{\mathbf{7a}}\)
7a (233)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7a}}&{}\approx{}&1.018702774091&{}:{}&-0.008977331564&{}:{}&-0.009725442528&,\\B^\prime_{\mathbf{7a}}&{}\approx{}&1.433384247027&{}:{}&3.293414795243&{}:{}&-3.726799042270&,\\C^\prime_{\mathbf{7a}}&{}\approx{}&0.902498713713&{}:{}&-2.165996912912&{}:{}&2.263498199199&, \\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.107613233147&{}:{}&-0.056682308091&{}:{}&-0.050930925055&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.018223135134&{}:{}&-0.017880554221&{}:{}&-0.000342580913&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.019782108871&{}:{}&-0.000381850666&{}:{}&-0.019400258205&, \\ A^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.172244979437&{}:{}&0.395758614525&{}:{}&0.431996406037&,\\B^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.999603466355&{}:{}&-0.008809018664&{}:{}&0.009205552309&,\\C^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.001111428293&{}:{}&0.008446071605&{}:{}&-0.009557499898&, \\ A^*_{\mathbf{7a}}&{}\approx{}&0.000000000000&{}:{}&0.563708941166&{}:{}&0.436291058834&,\\B^*_{\mathbf{7a}}&{}\approx{}&0.960600743902&{}:{}&0.000000000000&{}:{}&0.039399256098&,\\C^*_{\mathbf{7a}}&{}\approx{}&0.949673356358&{}:{}&0.050326643642&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7a}}}{B^\prime_{\mathbf{7a}}}{C^\prime_{\mathbf{7a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7a}}}{B^{\prime\prime}_{\mathbf{7a}}}{C^{\prime\prime}_{\mathbf{7a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7a}}}{B^{\prime\prime\prime}_{\mathbf{7a}}}{C^{\prime\prime\prime}_{\mathbf{7a}}}\)
\(\triangle{A^*_{\mathbf{7a}}}{B^*_{\mathbf{7a}}}{C^*_{\mathbf{7a}}}\)
7a (233)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7a}}}}&{}\approx{}&-0.014962219273&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7a}}}}&{}\approx{}&-5.733536988108&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7a}}}}&{}\approx{}&-3.609994854854&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.250000000000&{}:{}&0.600000000000&{}:{}&0.650000000000&,\\ A^\prime_{\mathbf{7a}}&{}\approx{}&1.018702774091&{}:{}&-0.008977331564&{}:{}&-0.009725442528&,\\B^\prime_{\mathbf{7a}}&{}\approx{}&1.433384247027&{}:{}&3.293414795243&{}:{}&-3.726799042270&,\\C^\prime_{\mathbf{7a}}&{}\approx{}&0.902498713713&{}:{}&-2.165996912912&{}:{}&2.263498199199&. \end{alignedat} \]
7a (233)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7a}}&{}\approx{}&1.000711398859&{}:{}&-0.000374709765&{}:{}&-0.000336689094&,\\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.107613233147&{}:{}&-0.056682308091&{}:{}&-0.050930925055&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.018223135134&{}:{}&-0.017880554221&{}:{}&-0.000342580913&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.019782108871&{}:{}&-0.000381850666&{}:{}&-0.019400258205&. \end{alignedat} \]
7a (233)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7a}}&{}\approx{}&0.982660079590&{}:{}&0.008290403207&{}:{}&0.009049517202&,\\ A^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.172244979437&{}:{}&0.395758614525&{}:{}&0.431996406037&,\\B^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.999603466355&{}:{}&-0.008809018664&{}:{}&0.009205552309&,\\C^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.001111428293&{}:{}&0.008446071605&{}:{}&-0.009557499898&. \end{alignedat} \]
7a (233)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7a}}&{}\approx{}&1.019299269712&{}:{}&-0.009297381757&{}:{}&-0.010001887955&,\\ A^\prime_{\mathbf{7a}}&{}\approx{}&1.018702774091&{}:{}&-0.008977331564&{}:{}&-0.009725442528&,\\B^\prime_{\mathbf{7a}}&{}\approx{}&1.433384247027&{}:{}&3.293414795243&{}:{}&-3.726799042270&,\\C^\prime_{\mathbf{7a}}&{}\approx{}&0.902498713713&{}:{}&-2.165996912912&{}:{}&2.263498199199&,\\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.107613233147&{}:{}&-0.056682308091&{}:{}&-0.050930925055&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.018223135134&{}:{}&-0.017880554221&{}:{}&-0.000342580913&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.019782108871&{}:{}&-0.000381850666&{}:{}&-0.019400258205&, \end{alignedat} \]
7a (233)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7a}}&{}\approx{}&0.914069378891&{}:{}&0.048439859439&{}:{}&0.037490761670&,\\ A^*_{\mathbf{7a}}&{}\approx{}&0.000000000000&{}:{}&0.563708941166&{}:{}&0.436291058834&,\\B^*_{\mathbf{7a}}&{}\approx{}&0.960600743902&{}:{}&0.000000000000&{}:{}&0.039399256098&,\\C^*_{\mathbf{7a}}&{}\approx{}&0.949673356358&{}:{}&0.050326643642&{}:{}&0.000000000000&. \end{alignedat} \]
7a (233)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7a}}&{}\approx{}&1.063325529863&{}:{}&-0.031876082646&{}:{}&-0.031449447216&,\\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.107613233147&{}:{}&-0.056682308091&{}:{}&-0.050930925055&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.018223135134&{}:{}&-0.017880554221&{}:{}&-0.000342580913&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.019782108871&{}:{}&-0.000381850666&{}:{}&-0.019400258205&,\\ A^*_{\mathbf{7a}}&{}\approx{}&0.000000000000&{}:{}&0.563708941166&{}:{}&0.436291058834&,\\B^*_{\mathbf{7a}}&{}\approx{}&0.960600743902&{}:{}&0.000000000000&{}:{}&0.039399256098&,\\C^*_{\mathbf{7a}}&{}\approx{}&0.949673356358&{}:{}&0.050326643642&{}:{}&0.000000000000&. \end{alignedat} \]
7a (233)