Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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5c(312)

Malfatti circles

5c (312)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.500000000000&{}:{}&2.000000000000&{}:{}&-2.500000000000&, \\ P_{\mathbf{5c}}&{}\approx{}&-0.128438083528&{}:{}&-0.038228759754&{}:{}&1.166666843281&, \\ P^-_{\mathbf{5c}}&{}\approx{}&-0.023239415330&{}:{}&0.093442780710&{}:{}&0.929796634620&, \\ P^+_{\mathbf{5c}}&{}\approx{}&-0.249245273290&{}:{}&-0.189436650952&{}:{}&1.438681924242&, \\ Q_{\mathbf{5c}}&{}\approx{}&-0.843511437593&{}:{}&0.375745287241&{}:{}&1.467766150353&, \\ I^\prime_{\mathbf{5c}}&{}\approx{}&-0.442293769655&{}:{}&-0.278630327865&{}:{}&1.720924097520&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{5c}}\)
\(P^-_{\mathbf{5c}}\)
\(P^+_{\mathbf{5c}}\)
\(Q_{\mathbf{5c}}\)
\(I^\prime_{\mathbf{5c}}\)
5c (312)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5c}}&{}\approx{}&-2.564755078384&{}:{}&-14.259020313537&{}:{}&17.823775391922&,\\B^\prime_{\mathbf{5c}}&{}\approx{}&-0.613130530351&{}:{}&0.591246313100&{}:{}&1.021884217251&,\\C^\prime_{\mathbf{5c}}&{}\approx{}&-0.137704095259&{}:{}&-0.183605460346&{}:{}&1.321309555605&, \\ A^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.224897952433&{}:{}&-0.041496587389&{}:{}&1.266394539822&,\\B^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.168558582526&{}:{}&-0.362542660834&{}:{}&1.531101243361&,\\C^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.718955126836&{}:{}&-0.213992314916&{}:{}&1.932947441751&, \\ A^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.104607008850&{}:{}&0.100873313667&{}:{}&1.003733695183&,\\B^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.029895487332&{}:{}&-0.166207044387&{}:{}&1.196102531719&,\\C^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.100297092599&{}:{}&0.403282057511&{}:{}&0.697015035088&, \\ A^*_{\mathbf{5c}}&{}\approx{}&0.000000000000&{}:{}&0.203820426377&{}:{}&0.796179573623&,\\B^*_{\mathbf{5c}}&{}\approx{}&-1.351229586822&{}:{}&0.000000000000&{}:{}&2.351229586822&,\\C^*_{\mathbf{5c}}&{}\approx{}&1.803275925284&{}:{}&-0.803275925284&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5c}}}{B^\prime_{\mathbf{5c}}}{C^\prime_{\mathbf{5c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5c}}}{B^{\prime\prime}_{\mathbf{5c}}}{C^{\prime\prime}_{\mathbf{5c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5c}}}{B^{\prime\prime\prime}_{\mathbf{5c}}}{C^{\prime\prime\prime}_{\mathbf{5c}}}\)
\(\triangle{A^*_{\mathbf{5c}}}{B^*_{\mathbf{5c}}}{C^*_{\mathbf{5c}}}\)
5c (312)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5c}}}}&{}\approx{}&-7.129510156769&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5c}}}}&{}\approx{}&-0.408753686900&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5c}}}}&{}\approx{}&-0.091802730173&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.500000000000&{}:{}&2.000000000000&{}:{}&-2.500000000000&,\\ A^\prime_{\mathbf{5c}}&{}\approx{}&-2.564755078384&{}:{}&-14.259020313537&{}:{}&17.823775391922&,\\B^\prime_{\mathbf{5c}}&{}\approx{}&-0.613130530351&{}:{}&0.591246313100&{}:{}&1.021884217251&,\\C^\prime_{\mathbf{5c}}&{}\approx{}&-0.137704095259&{}:{}&-0.183605460346&{}:{}&1.321309555605&. \end{alignedat} \]
5c (312)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5c}}&{}\approx{}&-0.128438083528&{}:{}&-0.038228759754&{}:{}&1.166666843281&,\\ A^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.224897952433&{}:{}&-0.041496587389&{}:{}&1.266394539822&,\\B^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.168558582526&{}:{}&-0.362542660834&{}:{}&1.531101243361&,\\C^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.718955126836&{}:{}&-0.213992314916&{}:{}&1.932947441751&. \end{alignedat} \]
5c (312)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5c}}&{}\approx{}&-0.023239415330&{}:{}&0.093442780710&{}:{}&0.929796634620&,\\ A^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.104607008850&{}:{}&0.100873313667&{}:{}&1.003733695183&,\\B^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.029895487332&{}:{}&-0.166207044387&{}:{}&1.196102531719&,\\C^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.100297092599&{}:{}&0.403282057511&{}:{}&0.697015035088&. \end{alignedat} \]
5c (312)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5c}}&{}\approx{}&-0.249245273290&{}:{}&-0.189436650952&{}:{}&1.438681924242&,\\ A^\prime_{\mathbf{5c}}&{}\approx{}&-2.564755078384&{}:{}&-14.259020313537&{}:{}&17.823775391922&,\\B^\prime_{\mathbf{5c}}&{}\approx{}&-0.613130530351&{}:{}&0.591246313100&{}:{}&1.021884217251&,\\C^\prime_{\mathbf{5c}}&{}\approx{}&-0.137704095259&{}:{}&-0.183605460346&{}:{}&1.321309555605&,\\ A^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.224897952433&{}:{}&-0.041496587389&{}:{}&1.266394539822&,\\B^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.168558582526&{}:{}&-0.362542660834&{}:{}&1.531101243361&,\\C^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.718955126836&{}:{}&-0.213992314916&{}:{}&1.932947441751&, \end{alignedat} \]
5c (312)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5c}}&{}\approx{}&-0.843511437593&{}:{}&0.375745287241&{}:{}&1.467766150353&,\\ A^*_{\mathbf{5c}}&{}\approx{}&0.000000000000&{}:{}&0.203820426377&{}:{}&0.796179573623&,\\B^*_{\mathbf{5c}}&{}\approx{}&-1.351229586822&{}:{}&0.000000000000&{}:{}&2.351229586822&,\\C^*_{\mathbf{5c}}&{}\approx{}&1.803275925284&{}:{}&-0.803275925284&{}:{}&0.000000000000&. \end{alignedat} \]
5c (312)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5c}}&{}\approx{}&-0.442293769655&{}:{}&-0.278630327865&{}:{}&1.720924097520&,\\ A^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.224897952433&{}:{}&-0.041496587389&{}:{}&1.266394539822&,\\B^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.168558582526&{}:{}&-0.362542660834&{}:{}&1.531101243361&,\\C^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.718955126836&{}:{}&-0.213992314916&{}:{}&1.932947441751&,\\ A^*_{\mathbf{5c}}&{}\approx{}&0.000000000000&{}:{}&0.203820426377&{}:{}&0.796179573623&,\\B^*_{\mathbf{5c}}&{}\approx{}&-1.351229586822&{}:{}&0.000000000000&{}:{}&2.351229586822&,\\C^*_{\mathbf{5c}}&{}\approx{}&1.803275925284&{}:{}&-0.803275925284&{}:{}&0.000000000000&. \end{alignedat} \]
5c (312)