Derousseau's Generalization of the Malfatti circles

Pythagorean Triangle U

\(C=90\degree\).   \(a:b:c=5:12:13\).


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1a(013)

Malfatti circles

1a (013)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.250000000000&{}:{}&0.600000000000&{}:{}&0.650000000000&, \\ P_{\mathbf{1a}}&{}\approx{}&-0.020008952338&{}:{}&0.949982617282&{}:{}&0.070026335057&, \\ P^-_{\mathbf{1a}}&{}\approx{}&0.006762042702&{}:{}&0.990720650378&{}:{}&0.002517306919&, \\ P^+_{\mathbf{1a}}&{}\approx{}&-0.041724549383&{}:{}&0.916937498497&{}:{}&0.124787050886&, \\ Q_{\mathbf{1a}}&{}\approx{}&0.063746890279&{}:{}&1.202915967926&{}:{}&-0.266662858205&, \\ I^\prime_{\mathbf{1a}}&{}\approx{}&-0.078795346806&{}:{}&0.841108160721&{}:{}&0.237687186085&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{1a}}\)
\(P^-_{\mathbf{1a}}\)
\(P^+_{\mathbf{1a}}\)
\(Q_{\mathbf{1a}}\)
\(I^\prime_{\mathbf{1a}}\)
1a (013)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1a}}&{}\approx{}&0.044410501982&{}:{}&0.458682959049&{}:{}&0.496906538969&,\\B^\prime_{\mathbf{1a}}&{}\approx{}&-0.028054161137&{}:{}&0.955113342180&{}:{}&0.072940818957&,\\C^\prime_{\mathbf{1a}}&{}\approx{}&-0.672718770973&{}:{}&1.614525050336&{}:{}&0.058193720637&, \\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.053862128333&{}:{}&0.981511682454&{}:{}&0.072350445879&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.114249398737&{}:{}&0.714405042035&{}:{}&0.399844356702&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.018778638842&{}:{}&0.891569942010&{}:{}&0.127208696832&, \\ A^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.030182445693&{}:{}&1.027571505002&{}:{}&0.002610940691&,\\B^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&0.085466254011&{}:{}&0.882717207399&{}:{}&0.031816538590&,\\C^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&0.006301158351&{}:{}&0.923195545183&{}:{}&0.070503296466&, \\ A^*_{\mathbf{1a}}&{}\approx{}&0.000000000000&{}:{}&1.284819196258&{}:{}&-0.284819196258&,\\B^*_{\mathbf{1a}}&{}\approx{}&-0.314154134496&{}:{}&0.000000000000&{}:{}&1.314154134496&,\\C^*_{\mathbf{1a}}&{}\approx{}&0.050326643642&{}:{}&0.949673356358&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1a}}}{B^\prime_{\mathbf{1a}}}{C^\prime_{\mathbf{1a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1a}}}{B^{\prime\prime}_{\mathbf{1a}}}{C^{\prime\prime}_{\mathbf{1a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1a}}}{B^{\prime\prime\prime}_{\mathbf{1a}}}{C^{\prime\prime\prime}_{\mathbf{1a}}}\)
\(\triangle{A^*_{\mathbf{1a}}}{B^*_{\mathbf{1a}}}{C^*_{\mathbf{1a}}}\)
1a (013)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1a}}}}&{}\approx{}&0.764471598414&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1a}}}}&{}\approx{}&0.112216644549&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1a}}}}&{}\approx{}&2.690875083894&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.250000000000&{}:{}&0.600000000000&{}:{}&0.650000000000&,\\ A^\prime_{\mathbf{1a}}&{}\approx{}&0.044410501982&{}:{}&0.458682959049&{}:{}&0.496906538969&,\\B^\prime_{\mathbf{1a}}&{}\approx{}&-0.028054161137&{}:{}&0.955113342180&{}:{}&0.072940818957&,\\C^\prime_{\mathbf{1a}}&{}\approx{}&-0.672718770973&{}:{}&1.614525050336&{}:{}&0.058193720637&. \end{alignedat} \]
1a (013)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1a}}&{}\approx{}&-0.020008952338&{}:{}&0.949982617282&{}:{}&0.070026335057&,\\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.053862128333&{}:{}&0.981511682454&{}:{}&0.072350445879&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.114249398737&{}:{}&0.714405042035&{}:{}&0.399844356702&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.018778638842&{}:{}&0.891569942010&{}:{}&0.127208696832&. \end{alignedat} \]
1a (013)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1a}}&{}\approx{}&0.006762042702&{}:{}&0.990720650378&{}:{}&0.002517306919&,\\ A^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.030182445693&{}:{}&1.027571505002&{}:{}&0.002610940691&,\\B^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&0.085466254011&{}:{}&0.882717207399&{}:{}&0.031816538590&,\\C^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&0.006301158351&{}:{}&0.923195545183&{}:{}&0.070503296466&. \end{alignedat} \]
1a (013)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1a}}&{}\approx{}&-0.041724549383&{}:{}&0.916937498497&{}:{}&0.124787050886&,\\ A^\prime_{\mathbf{1a}}&{}\approx{}&0.044410501982&{}:{}&0.458682959049&{}:{}&0.496906538969&,\\B^\prime_{\mathbf{1a}}&{}\approx{}&-0.028054161137&{}:{}&0.955113342180&{}:{}&0.072940818957&,\\C^\prime_{\mathbf{1a}}&{}\approx{}&-0.672718770973&{}:{}&1.614525050336&{}:{}&0.058193720637&,\\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.053862128333&{}:{}&0.981511682454&{}:{}&0.072350445879&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.114249398737&{}:{}&0.714405042035&{}:{}&0.399844356702&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.018778638842&{}:{}&0.891569942010&{}:{}&0.127208696832&, \end{alignedat} \]
1a (013)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1a}}&{}\approx{}&0.063746890279&{}:{}&1.202915967926&{}:{}&-0.266662858205&,\\ A^*_{\mathbf{1a}}&{}\approx{}&0.000000000000&{}:{}&1.284819196258&{}:{}&-0.284819196258&,\\B^*_{\mathbf{1a}}&{}\approx{}&-0.314154134496&{}:{}&0.000000000000&{}:{}&1.314154134496&,\\C^*_{\mathbf{1a}}&{}\approx{}&0.050326643642&{}:{}&0.949673356358&{}:{}&0.000000000000&. \end{alignedat} \]
1a (013)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1a}}&{}\approx{}&-0.078795346806&{}:{}&0.841108160721&{}:{}&0.237687186085&,\\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.053862128333&{}:{}&0.981511682454&{}:{}&0.072350445879&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.114249398737&{}:{}&0.714405042035&{}:{}&0.399844356702&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.018778638842&{}:{}&0.891569942010&{}:{}&0.127208696832&,\\ A^*_{\mathbf{1a}}&{}\approx{}&0.000000000000&{}:{}&1.284819196258&{}:{}&-0.284819196258&,\\B^*_{\mathbf{1a}}&{}\approx{}&-0.314154134496&{}:{}&0.000000000000&{}:{}&1.314154134496&,\\C^*_{\mathbf{1a}}&{}\approx{}&0.050326643642&{}:{}&0.949673356358&{}:{}&0.000000000000&. \end{alignedat} \]
1a (013)