Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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6a(231)

Malfatti circles

6a (231)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.500000000000&{}:{}&0.666666666667&{}:{}&0.833333333333&, \\ P_{\mathbf{6a}}&{}\approx{}&1.013167773897&{}:{}&-0.011641873393&{}:{}&-0.001525900504&, \\ P^-_{\mathbf{6a}}&{}\approx{}&0.934151405280&{}:{}&0.023778837094&{}:{}&0.042069757626&, \\ P^+_{\mathbf{6a}}&{}\approx{}&1.101398850039&{}:{}&-0.051193265829&{}:{}&-0.050205584211&, \\ Q_{\mathbf{6a}}&{}\approx{}&1.242781246185&{}:{}&-0.326318028463&{}:{}&0.083536782278&, \\ I^\prime_{\mathbf{6a}}&{}\approx{}&1.210475104788&{}:{}&-0.149829035022&{}:{}&-0.060646069766&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{6a}}\)
\(P^-_{\mathbf{6a}}\)
\(P^+_{\mathbf{6a}}\)
\(Q_{\mathbf{6a}}\)
\(I^\prime_{\mathbf{6a}}\)
6a (231)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6a}}&{}\approx{}&1.091524521636&{}:{}&-0.040677565172&{}:{}&-0.050846956465&,\\B^\prime_{\mathbf{6a}}&{}\approx{}&4.097262605646&{}:{}&3.731508403764&{}:{}&-6.828771009409&,\\C^\prime_{\mathbf{6a}}&{}\approx{}&0.782507527261&{}:{}&-1.043343369682&{}:{}&1.260835842420&, \\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.314053220853&{}:{}&-0.277660283693&{}:{}&-0.036392937160&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.079928750260&{}:{}&-0.078302303124&{}:{}&-0.001626447136&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.113739800778&{}:{}&-0.012797503126&{}:{}&-0.100942297652&, \\ A^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.283928910755&{}:{}&0.258583161132&{}:{}&0.457487928113&,\\B^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.992291362309&{}:{}&-0.036979468404&{}:{}&0.044688106094&,\\C^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.021584461791&{}:{}&0.026004446771&{}:{}&-0.047588908562&, \\ A^*_{\mathbf{6a}}&{}\approx{}&0.000000000000&{}:{}&1.344082517041&{}:{}&-0.344082517041&,\\B^*_{\mathbf{6a}}&{}\approx{}&0.937016024449&{}:{}&0.000000000000&{}:{}&0.062983975551&,\\C^*_{\mathbf{6a}}&{}\approx{}&1.356062329784&{}:{}&-0.356062329784&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6a}}}{B^\prime_{\mathbf{6a}}}{C^\prime_{\mathbf{6a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6a}}}{B^{\prime\prime}_{\mathbf{6a}}}{C^{\prime\prime}_{\mathbf{6a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6a}}}{B^{\prime\prime\prime}_{\mathbf{6a}}}{C^{\prime\prime\prime}_{\mathbf{6a}}}\)
\(\triangle{A^*_{\mathbf{6a}}}{B^*_{\mathbf{6a}}}{C^*_{\mathbf{6a}}}\)
6a (231)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6a}}}}&{}\approx{}&-0.061016347758&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6a}}}}&{}\approx{}&-8.194525211291&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6a}}}}&{}\approx{}&-1.565015054523&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.500000000000&{}:{}&0.666666666667&{}:{}&0.833333333333&,\\ A^\prime_{\mathbf{6a}}&{}\approx{}&1.091524521636&{}:{}&-0.040677565172&{}:{}&-0.050846956465&,\\B^\prime_{\mathbf{6a}}&{}\approx{}&4.097262605646&{}:{}&3.731508403764&{}:{}&-6.828771009409&,\\C^\prime_{\mathbf{6a}}&{}\approx{}&0.782507527261&{}:{}&-1.043343369682&{}:{}&1.260835842420&. \end{alignedat} \]
6a (231)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6a}}&{}\approx{}&1.013167773897&{}:{}&-0.011641873393&{}:{}&-0.001525900504&,\\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.314053220853&{}:{}&-0.277660283693&{}:{}&-0.036392937160&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.079928750260&{}:{}&-0.078302303124&{}:{}&-0.001626447136&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.113739800778&{}:{}&-0.012797503126&{}:{}&-0.100942297652&. \end{alignedat} \]
6a (231)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6a}}&{}\approx{}&0.934151405280&{}:{}&0.023778837094&{}:{}&0.042069757626&,\\ A^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.283928910755&{}:{}&0.258583161132&{}:{}&0.457487928113&,\\B^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.992291362309&{}:{}&-0.036979468404&{}:{}&0.044688106094&,\\C^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.021584461791&{}:{}&0.026004446771&{}:{}&-0.047588908562&. \end{alignedat} \]
6a (231)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6a}}&{}\approx{}&1.101398850039&{}:{}&-0.051193265829&{}:{}&-0.050205584211&,\\ A^\prime_{\mathbf{6a}}&{}\approx{}&1.091524521636&{}:{}&-0.040677565172&{}:{}&-0.050846956465&,\\B^\prime_{\mathbf{6a}}&{}\approx{}&4.097262605646&{}:{}&3.731508403764&{}:{}&-6.828771009409&,\\C^\prime_{\mathbf{6a}}&{}\approx{}&0.782507527261&{}:{}&-1.043343369682&{}:{}&1.260835842420&,\\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.314053220853&{}:{}&-0.277660283693&{}:{}&-0.036392937160&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.079928750260&{}:{}&-0.078302303124&{}:{}&-0.001626447136&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.113739800778&{}:{}&-0.012797503126&{}:{}&-0.100942297652&, \end{alignedat} \]
6a (231)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6a}}&{}\approx{}&1.242781246185&{}:{}&-0.326318028463&{}:{}&0.083536782278&,\\ A^*_{\mathbf{6a}}&{}\approx{}&0.000000000000&{}:{}&1.344082517041&{}:{}&-0.344082517041&,\\B^*_{\mathbf{6a}}&{}\approx{}&0.937016024449&{}:{}&0.000000000000&{}:{}&0.062983975551&,\\C^*_{\mathbf{6a}}&{}\approx{}&1.356062329784&{}:{}&-0.356062329784&{}:{}&0.000000000000&. \end{alignedat} \]
6a (231)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6a}}&{}\approx{}&1.210475104788&{}:{}&-0.149829035022&{}:{}&-0.060646069766&,\\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.314053220853&{}:{}&-0.277660283693&{}:{}&-0.036392937160&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.079928750260&{}:{}&-0.078302303124&{}:{}&-0.001626447136&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.113739800778&{}:{}&-0.012797503126&{}:{}&-0.100942297652&,\\ A^*_{\mathbf{6a}}&{}\approx{}&0.000000000000&{}:{}&1.344082517041&{}:{}&-0.344082517041&,\\B^*_{\mathbf{6a}}&{}\approx{}&0.937016024449&{}:{}&0.000000000000&{}:{}&0.062983975551&,\\C^*_{\mathbf{6a}}&{}\approx{}&1.356062329784&{}:{}&-0.356062329784&{}:{}&0.000000000000&. \end{alignedat} \]
6a (231)