Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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7c(332)

Malfatti circles

7c (332)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.500000000000&{}:{}&2.000000000000&{}:{}&-2.500000000000&, \\ P_{\mathbf{7c}}&{}\approx{}&-0.031690292878&{}:{}&-0.034951178024&{}:{}&1.066641470902&, \\ P^-_{\mathbf{7c}}&{}\approx{}&0.016955824486&{}:{}&0.029678384686&{}:{}&0.953365790828&, \\ P^+_{\mathbf{7c}}&{}\approx{}&-0.083635975130&{}:{}&-0.103964429452&{}:{}&1.187600404583&, \\ Q_{\mathbf{7c}}&{}\approx{}&0.192048473131&{}:{}&0.164677024698&{}:{}&0.643274502171&, \\ I^\prime_{\mathbf{7c}}&{}\approx{}&-0.189496231731&{}:{}&-0.229793494234&{}:{}&1.419289725966&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{7c}}\)
\(P^-_{\mathbf{7c}}\)
\(P^+_{\mathbf{7c}}\)
\(Q_{\mathbf{7c}}\)
\(I^\prime_{\mathbf{7c}}\)
7c (332)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7c}}&{}\approx{}&0.516383751700&{}:{}&-1.934464993201&{}:{}&2.418081241501&,\\B^\prime_{\mathbf{7c}}&{}\approx{}&-0.775408190519&{}:{}&0.483061206321&{}:{}&1.292346984198&,\\C^\prime_{\mathbf{7c}}&{}\approx{}&-0.056565265175&{}:{}&-0.075420353567&{}:{}&1.131985618742&, \\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.105438868549&{}:{}&-0.037449601839&{}:{}&1.142888470388&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.035065522666&{}:{}&-0.145180454184&{}:{}&1.180245976850&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.325475374362&{}:{}&-0.358966318031&{}:{}&1.684441692393&, \\ A^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.050929452976&{}:{}&0.031727860619&{}:{}&1.019201592357&,\\B^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&0.018698484051&{}:{}&-0.070047833038&{}:{}&1.051349348987&,\\C^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&0.134534147826&{}:{}&0.235479919949&{}:{}&0.629985932225&, \\ A^*_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&0.203820426377&{}:{}&0.796179573623&,\\B^*_{\mathbf{7c}}&{}\approx{}&0.229909243262&{}:{}&0.000000000000&{}:{}&0.770090756738&,\\C^*_{\mathbf{7c}}&{}\approx{}&0.538364861216&{}:{}&0.461635138784&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7c}}}{B^\prime_{\mathbf{7c}}}{C^\prime_{\mathbf{7c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7c}}}{B^{\prime\prime}_{\mathbf{7c}}}{C^{\prime\prime}_{\mathbf{7c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7c}}}{B^{\prime\prime\prime}_{\mathbf{7c}}}{C^{\prime\prime\prime}_{\mathbf{7c}}}\)
\(\triangle{A^*_{\mathbf{7c}}}{B^*_{\mathbf{7c}}}{C^*_{\mathbf{7c}}}\)
7c (332)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7c}}}}&{}\approx{}&-0.967232496600&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7c}}}}&{}\approx{}&-0.516938793679&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7c}}}}&{}\approx{}&-0.037710176784&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.500000000000&{}:{}&2.000000000000&{}:{}&-2.500000000000&,\\ A^\prime_{\mathbf{7c}}&{}\approx{}&0.516383751700&{}:{}&-1.934464993201&{}:{}&2.418081241501&,\\B^\prime_{\mathbf{7c}}&{}\approx{}&-0.775408190519&{}:{}&0.483061206321&{}:{}&1.292346984198&,\\C^\prime_{\mathbf{7c}}&{}\approx{}&-0.056565265175&{}:{}&-0.075420353567&{}:{}&1.131985618742&. \end{alignedat} \]
7c (332)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7c}}&{}\approx{}&-0.031690292878&{}:{}&-0.034951178024&{}:{}&1.066641470902&,\\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.105438868549&{}:{}&-0.037449601839&{}:{}&1.142888470388&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.035065522666&{}:{}&-0.145180454184&{}:{}&1.180245976850&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.325475374362&{}:{}&-0.358966318031&{}:{}&1.684441692393&. \end{alignedat} \]
7c (332)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7c}}&{}\approx{}&0.016955824486&{}:{}&0.029678384686&{}:{}&0.953365790828&,\\ A^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.050929452976&{}:{}&0.031727860619&{}:{}&1.019201592357&,\\B^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&0.018698484051&{}:{}&-0.070047833038&{}:{}&1.051349348987&,\\C^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&0.134534147826&{}:{}&0.235479919949&{}:{}&0.629985932225&. \end{alignedat} \]
7c (332)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7c}}&{}\approx{}&-0.083635975130&{}:{}&-0.103964429452&{}:{}&1.187600404583&,\\ A^\prime_{\mathbf{7c}}&{}\approx{}&0.516383751700&{}:{}&-1.934464993201&{}:{}&2.418081241501&,\\B^\prime_{\mathbf{7c}}&{}\approx{}&-0.775408190519&{}:{}&0.483061206321&{}:{}&1.292346984198&,\\C^\prime_{\mathbf{7c}}&{}\approx{}&-0.056565265175&{}:{}&-0.075420353567&{}:{}&1.131985618742&,\\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.105438868549&{}:{}&-0.037449601839&{}:{}&1.142888470388&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.035065522666&{}:{}&-0.145180454184&{}:{}&1.180245976850&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.325475374362&{}:{}&-0.358966318031&{}:{}&1.684441692393&, \end{alignedat} \]
7c (332)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7c}}&{}\approx{}&0.192048473131&{}:{}&0.164677024698&{}:{}&0.643274502171&,\\ A^*_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&0.203820426377&{}:{}&0.796179573623&,\\B^*_{\mathbf{7c}}&{}\approx{}&0.229909243262&{}:{}&0.000000000000&{}:{}&0.770090756738&,\\C^*_{\mathbf{7c}}&{}\approx{}&0.538364861216&{}:{}&0.461635138784&{}:{}&0.000000000000&. \end{alignedat} \]
7c (332)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7c}}&{}\approx{}&-0.189496231731&{}:{}&-0.229793494234&{}:{}&1.419289725966&,\\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.105438868549&{}:{}&-0.037449601839&{}:{}&1.142888470388&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.035065522666&{}:{}&-0.145180454184&{}:{}&1.180245976850&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.325475374362&{}:{}&-0.358966318031&{}:{}&1.684441692393&,\\ A^*_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&0.203820426377&{}:{}&0.796179573623&,\\B^*_{\mathbf{7c}}&{}\approx{}&0.229909243262&{}:{}&0.000000000000&{}:{}&0.770090756738&,\\C^*_{\mathbf{7c}}&{}\approx{}&0.538364861216&{}:{}&0.461635138784&{}:{}&0.000000000000&. \end{alignedat} \]
7c (332)