Derousseau's Generalization of the Malfatti circles

Pythagorean Triangle U

\(C=90\degree\).   \(a:b:c=5:12:13\).


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3a(033)

Malfatti circles

3a (033)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.250000000000&{}:{}&0.600000000000&{}:{}&0.650000000000&, \\ P_{\mathbf{3a}}&{}\approx{}&-0.006047179017&{}:{}&0.287107233441&{}:{}&0.718939945576&, \\ P^-_{\mathbf{3a}}&{}\approx{}&0.048837113559&{}:{}&0.216712892127&{}:{}&0.734449994314&, \\ P^+_{\mathbf{3a}}&{}\approx{}&-0.043899505083&{}:{}&0.335656452818&{}:{}&0.708243052266&, \\ Q_{\mathbf{3a}}&{}\approx{}&0.022598165071&{}:{}&0.426431681468&{}:{}&0.550970153461&, \\ I^\prime_{\mathbf{3a}}&{}\approx{}&-0.036689010756&{}:{}&0.391640212356&{}:{}&0.645048798400&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{3a}}\)
\(P^-_{\mathbf{3a}}\)
\(P^+_{\mathbf{3a}}\)
\(Q_{\mathbf{3a}}\)
\(I^\prime_{\mathbf{3a}}\)
3a (033)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3a}}&{}\approx{}&0.147963892575&{}:{}&0.408977331564&{}:{}&0.443058775861&,\\B^\prime_{\mathbf{3a}}&{}\approx{}&-0.183384247027&{}:{}&0.706585204757&{}:{}&0.476799042270&,\\C^\prime_{\mathbf{3a}}&{}\approx{}&-0.069165380380&{}:{}&0.165996912912&{}:{}&0.903168467468&, \\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.100446326780&{}:{}&0.314047001991&{}:{}&0.786399324789&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.006479461211&{}:{}&0.236146140955&{}:{}&0.770333320256&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.011633273663&{}:{}&0.552323158913&{}:{}&0.459310114750&, \\ A^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.062849082309&{}:{}&0.242159467958&{}:{}&0.820689614351&,\\B^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.053183562326&{}:{}&0.147001211069&{}:{}&0.799815226606&,\\C^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.118598159291&{}:{}&0.526274962384&{}:{}&0.355126878325&, \\ A^*_{\mathbf{3a}}&{}\approx{}&0.000000000000&{}:{}&0.436291058834&{}:{}&0.563708941166&,\\B^*_{\mathbf{3a}}&{}\approx{}&0.039399256098&{}:{}&0.000000000000&{}:{}&0.960600743902&,\\C^*_{\mathbf{3a}}&{}\approx{}&0.050326643642&{}:{}&0.949673356358&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3a}}}{B^\prime_{\mathbf{3a}}}{C^\prime_{\mathbf{3a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3a}}}{B^{\prime\prime}_{\mathbf{3a}}}{C^{\prime\prime}_{\mathbf{3a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3a}}}{B^{\prime\prime\prime}_{\mathbf{3a}}}{C^{\prime\prime\prime}_{\mathbf{3a}}}\)
\(\triangle{A^*_{\mathbf{3a}}}{B^*_{\mathbf{3a}}}{C^*_{\mathbf{3a}}}\)
3a (033)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3a}}}}&{}\approx{}&0.681628885940&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3a}}}}&{}\approx{}&0.733536988108&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3a}}}}&{}\approx{}&0.276661521521&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.250000000000&{}:{}&0.600000000000&{}:{}&0.650000000000&,\\ A^\prime_{\mathbf{3a}}&{}\approx{}&0.147963892575&{}:{}&0.408977331564&{}:{}&0.443058775861&,\\B^\prime_{\mathbf{3a}}&{}\approx{}&-0.183384247027&{}:{}&0.706585204757&{}:{}&0.476799042270&,\\C^\prime_{\mathbf{3a}}&{}\approx{}&-0.069165380380&{}:{}&0.165996912912&{}:{}&0.903168467468&. \end{alignedat} \]
3a (033)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3a}}&{}\approx{}&-0.006047179017&{}:{}&0.287107233441&{}:{}&0.718939945576&,\\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.100446326780&{}:{}&0.314047001991&{}:{}&0.786399324789&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.006479461211&{}:{}&0.236146140955&{}:{}&0.770333320256&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.011633273663&{}:{}&0.552323158913&{}:{}&0.459310114750&. \end{alignedat} \]
3a (033)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3a}}&{}\approx{}&0.048837113559&{}:{}&0.216712892127&{}:{}&0.734449994314&,\\ A^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.062849082309&{}:{}&0.242159467958&{}:{}&0.820689614351&,\\B^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.053183562326&{}:{}&0.147001211069&{}:{}&0.799815226606&,\\C^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.118598159291&{}:{}&0.526274962384&{}:{}&0.355126878325&. \end{alignedat} \]
3a (033)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3a}}&{}\approx{}&-0.043899505083&{}:{}&0.335656452818&{}:{}&0.708243052266&,\\ A^\prime_{\mathbf{3a}}&{}\approx{}&0.147963892575&{}:{}&0.408977331564&{}:{}&0.443058775861&,\\B^\prime_{\mathbf{3a}}&{}\approx{}&-0.183384247027&{}:{}&0.706585204757&{}:{}&0.476799042270&,\\C^\prime_{\mathbf{3a}}&{}\approx{}&-0.069165380380&{}:{}&0.165996912912&{}:{}&0.903168467468&,\\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.100446326780&{}:{}&0.314047001991&{}:{}&0.786399324789&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.006479461211&{}:{}&0.236146140955&{}:{}&0.770333320256&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.011633273663&{}:{}&0.552323158913&{}:{}&0.459310114750&, \end{alignedat} \]
3a (033)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3a}}&{}\approx{}&0.022598165071&{}:{}&0.426431681468&{}:{}&0.550970153461&,\\ A^*_{\mathbf{3a}}&{}\approx{}&0.000000000000&{}:{}&0.436291058834&{}:{}&0.563708941166&,\\B^*_{\mathbf{3a}}&{}\approx{}&0.039399256098&{}:{}&0.000000000000&{}:{}&0.960600743902&,\\C^*_{\mathbf{3a}}&{}\approx{}&0.050326643642&{}:{}&0.949673356358&{}:{}&0.000000000000&. \end{alignedat} \]
3a (033)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3a}}&{}\approx{}&-0.036689010756&{}:{}&0.391640212356&{}:{}&0.645048798400&,\\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.100446326780&{}:{}&0.314047001991&{}:{}&0.786399324789&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.006479461211&{}:{}&0.236146140955&{}:{}&0.770333320256&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.011633273663&{}:{}&0.552323158913&{}:{}&0.459310114750&,\\ A^*_{\mathbf{3a}}&{}\approx{}&0.000000000000&{}:{}&0.436291058834&{}:{}&0.563708941166&,\\B^*_{\mathbf{3a}}&{}\approx{}&0.039399256098&{}:{}&0.000000000000&{}:{}&0.960600743902&,\\C^*_{\mathbf{3a}}&{}\approx{}&0.050326643642&{}:{}&0.949673356358&{}:{}&0.000000000000&. \end{alignedat} \]
3a (033)