Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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6c(330)

Malfatti circles

6c (330)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.500000000000&{}:{}&2.000000000000&{}:{}&-2.500000000000&, \\ P_{\mathbf{6c}}&{}\approx{}&0.360661147201&{}:{}&0.735777177453&{}:{}&-0.096438324654&, \\ P^-_{\mathbf{6c}}&{}\approx{}&0.251941929525&{}:{}&0.615141147269&{}:{}&0.132916923205&, \\ P^+_{\mathbf{6c}}&{}\approx{}&0.451956918310&{}:{}&0.837079961216&{}:{}&-0.289036879526&, \\ Q_{\mathbf{6c}}&{}\approx{}&0.405718824309&{}:{}&0.473154533074&{}:{}&0.121126642617&, \\ I^\prime_{\mathbf{6c}}&{}\approx{}&0.504617259636&{}:{}&0.832252187871&{}:{}&-0.336869447506&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{6c}}\)
\(P^-_{\mathbf{6c}}\)
\(P^+_{\mathbf{6c}}\)
\(Q_{\mathbf{6c}}\)
\(I^\prime_{\mathbf{6c}}\)
6c (330)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6c}}&{}\approx{}&1.233616248300&{}:{}&0.934464993201&{}:{}&-1.168081241501&,\\B^\prime_{\mathbf{6c}}&{}\approx{}&0.275408190519&{}:{}&1.183605460346&{}:{}&-0.459013650865&,\\C^\prime_{\mathbf{6c}}&{}\approx{}&0.306565265175&{}:{}&0.408753686900&{}:{}&0.284681047924&, \\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.290152700718&{}:{}&0.816921167863&{}:{}&-0.107073868581&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.588675328794&{}:{}&0.568732428917&{}:{}&-0.157407757711&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.545724679713&{}:{}&1.113321375540&{}:{}&-0.659046055253&, \\ A^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.160610175781&{}:{}&0.690244835068&{}:{}&0.149144989151&,\\B^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.437623309955&{}:{}&0.331499900334&{}:{}&0.230876789710&,\\C^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.400846066979&{}:{}&0.978705330964&{}:{}&-0.379551397943&, \\ A^*_{\mathbf{6c}}&{}\approx{}&0.000000000000&{}:{}&0.796179573623&{}:{}&0.203820426377&,\\B^*_{\mathbf{6c}}&{}\approx{}&0.770090756738&{}:{}&0.000000000000&{}:{}&0.229909243262&,\\C^*_{\mathbf{6c}}&{}\approx{}&0.461635138784&{}:{}&0.538364861216&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6c}}}{B^\prime_{\mathbf{6c}}}{C^\prime_{\mathbf{6c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6c}}}{B^{\prime\prime}_{\mathbf{6c}}}{C^{\prime\prime}_{\mathbf{6c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6c}}}{B^{\prime\prime\prime}_{\mathbf{6c}}}{C^{\prime\prime\prime}_{\mathbf{6c}}}\)
\(\triangle{A^*_{\mathbf{6c}}}{B^*_{\mathbf{6c}}}{C^*_{\mathbf{6c}}}\)
6c (330)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6c}}}}&{}\approx{}&0.467232496600&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6c}}}}&{}\approx{}&0.183605460346&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6c}}}}&{}\approx{}&0.204376843450&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.500000000000&{}:{}&2.000000000000&{}:{}&-2.500000000000&,\\ A^\prime_{\mathbf{6c}}&{}\approx{}&1.233616248300&{}:{}&0.934464993201&{}:{}&-1.168081241501&,\\B^\prime_{\mathbf{6c}}&{}\approx{}&0.275408190519&{}:{}&1.183605460346&{}:{}&-0.459013650865&,\\C^\prime_{\mathbf{6c}}&{}\approx{}&0.306565265175&{}:{}&0.408753686900&{}:{}&0.284681047924&. \end{alignedat} \]
6c (330)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6c}}&{}\approx{}&0.360661147201&{}:{}&0.735777177453&{}:{}&-0.096438324654&,\\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.290152700718&{}:{}&0.816921167863&{}:{}&-0.107073868581&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.588675328794&{}:{}&0.568732428917&{}:{}&-0.157407757711&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.545724679713&{}:{}&1.113321375540&{}:{}&-0.659046055253&. \end{alignedat} \]
6c (330)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6c}}&{}\approx{}&0.251941929525&{}:{}&0.615141147269&{}:{}&0.132916923205&,\\ A^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.160610175781&{}:{}&0.690244835068&{}:{}&0.149144989151&,\\B^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.437623309955&{}:{}&0.331499900334&{}:{}&0.230876789710&,\\C^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.400846066979&{}:{}&0.978705330964&{}:{}&-0.379551397943&. \end{alignedat} \]
6c (330)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6c}}&{}\approx{}&0.451956918310&{}:{}&0.837079961216&{}:{}&-0.289036879526&,\\ A^\prime_{\mathbf{6c}}&{}\approx{}&1.233616248300&{}:{}&0.934464993201&{}:{}&-1.168081241501&,\\B^\prime_{\mathbf{6c}}&{}\approx{}&0.275408190519&{}:{}&1.183605460346&{}:{}&-0.459013650865&,\\C^\prime_{\mathbf{6c}}&{}\approx{}&0.306565265175&{}:{}&0.408753686900&{}:{}&0.284681047924&,\\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.290152700718&{}:{}&0.816921167863&{}:{}&-0.107073868581&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.588675328794&{}:{}&0.568732428917&{}:{}&-0.157407757711&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.545724679713&{}:{}&1.113321375540&{}:{}&-0.659046055253&, \end{alignedat} \]
6c (330)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6c}}&{}\approx{}&0.405718824309&{}:{}&0.473154533074&{}:{}&0.121126642617&,\\ A^*_{\mathbf{6c}}&{}\approx{}&0.000000000000&{}:{}&0.796179573623&{}:{}&0.203820426377&,\\B^*_{\mathbf{6c}}&{}\approx{}&0.770090756738&{}:{}&0.000000000000&{}:{}&0.229909243262&,\\C^*_{\mathbf{6c}}&{}\approx{}&0.461635138784&{}:{}&0.538364861216&{}:{}&0.000000000000&. \end{alignedat} \]
6c (330)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6c}}&{}\approx{}&0.504617259636&{}:{}&0.832252187871&{}:{}&-0.336869447506&,\\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.290152700718&{}:{}&0.816921167863&{}:{}&-0.107073868581&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.588675328794&{}:{}&0.568732428917&{}:{}&-0.157407757711&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.545724679713&{}:{}&1.113321375540&{}:{}&-0.659046055253&,\\ A^*_{\mathbf{6c}}&{}\approx{}&0.000000000000&{}:{}&0.796179573623&{}:{}&0.203820426377&,\\B^*_{\mathbf{6c}}&{}\approx{}&0.770090756738&{}:{}&0.000000000000&{}:{}&0.229909243262&,\\C^*_{\mathbf{6c}}&{}\approx{}&0.461635138784&{}:{}&0.538364861216&{}:{}&0.000000000000&. \end{alignedat} \]
6c (330)