Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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4c(310)

Malfatti circles

4c (310)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.500000000000&{}:{}&2.000000000000&{}:{}&-2.500000000000&, \\ P_{\mathbf{4c}}&{}\approx{}&0.132125334504&{}:{}&0.998785493743&{}:{}&-0.130910828247&, \\ P^-_{\mathbf{4c}}&{}\approx{}&0.042597657552&{}:{}&0.933255799293&{}:{}&0.024146543155&, \\ P^+_{\mathbf{4c}}&{}\approx{}&0.211290286782&{}:{}&1.056730202134&{}:{}&-0.268020488916&, \\ Q_{\mathbf{4c}}&{}\approx{}&-0.549573490654&{}:{}&1.233738761087&{}:{}&0.315834729567&, \\ I^\prime_{\mathbf{4c}}&{}\approx{}&0.346053821241&{}:{}&1.098641704032&{}:{}&-0.444695525273&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{4c}}\)
\(P^-_{\mathbf{4c}}\)
\(P^+_{\mathbf{4c}}\)
\(Q_{\mathbf{4c}}\)
\(I^\prime_{\mathbf{4c}}\)
4c (310)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4c}}&{}\approx{}&4.314755078384&{}:{}&13.259020313537&{}:{}&-16.573775391922&,\\B^\prime_{\mathbf{4c}}&{}\approx{}&0.113130530351&{}:{}&1.075420353567&{}:{}&-0.188550883918&,\\C^\prime_{\mathbf{4c}}&{}\approx{}&0.387704095259&{}:{}&0.516938793679&{}:{}&0.095357111061&, \\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.175152306462&{}:{}&0.949268302909&{}:{}&-0.124420609371&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.535065522666&{}:{}&0.995081636481&{}:{}&-0.530147159147&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.160392468186&{}:{}&1.212467473636&{}:{}&-0.372859941821&, \\ A^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.093006238989&{}:{}&0.884118567354&{}:{}&0.022875193657&,\\B^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.215526532702&{}:{}&0.662301943747&{}:{}&0.122171523550&,\\C^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.052445446895&{}:{}&1.149007252360&{}:{}&-0.201452699256&, \\ A^*_{\mathbf{4c}}&{}\approx{}&0.000000000000&{}:{}&0.796179573623&{}:{}&0.203820426377&,\\B^*_{\mathbf{4c}}&{}\approx{}&2.351229586822&{}:{}&0.000000000000&{}:{}&-1.351229586822&,\\C^*_{\mathbf{4c}}&{}\approx{}&-0.803275925284&{}:{}&1.803275925284&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4c}}}{B^\prime_{\mathbf{4c}}}{C^\prime_{\mathbf{4c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4c}}}{B^{\prime\prime}_{\mathbf{4c}}}{C^{\prime\prime}_{\mathbf{4c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4c}}}{B^{\prime\prime\prime}_{\mathbf{4c}}}{C^{\prime\prime\prime}_{\mathbf{4c}}}\)
\(\triangle{A^*_{\mathbf{4c}}}{B^*_{\mathbf{4c}}}{C^*_{\mathbf{4c}}}\)
4c (310)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4c}}}}&{}\approx{}&6.629510156769&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4c}}}}&{}\approx{}&0.075420353567&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4c}}}}&{}\approx{}&0.258469396840&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.500000000000&{}:{}&2.000000000000&{}:{}&-2.500000000000&,\\ A^\prime_{\mathbf{4c}}&{}\approx{}&4.314755078384&{}:{}&13.259020313537&{}:{}&-16.573775391922&,\\B^\prime_{\mathbf{4c}}&{}\approx{}&0.113130530351&{}:{}&1.075420353567&{}:{}&-0.188550883918&,\\C^\prime_{\mathbf{4c}}&{}\approx{}&0.387704095259&{}:{}&0.516938793679&{}:{}&0.095357111061&. \end{alignedat} \]
4c (310)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4c}}&{}\approx{}&0.132125334504&{}:{}&0.998785493743&{}:{}&-0.130910828247&,\\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.175152306462&{}:{}&0.949268302909&{}:{}&-0.124420609371&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.535065522666&{}:{}&0.995081636481&{}:{}&-0.530147159147&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.160392468186&{}:{}&1.212467473636&{}:{}&-0.372859941821&. \end{alignedat} \]
4c (310)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4c}}&{}\approx{}&0.042597657552&{}:{}&0.933255799293&{}:{}&0.024146543155&,\\ A^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.093006238989&{}:{}&0.884118567354&{}:{}&0.022875193657&,\\B^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.215526532702&{}:{}&0.662301943747&{}:{}&0.122171523550&,\\C^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.052445446895&{}:{}&1.149007252360&{}:{}&-0.201452699256&. \end{alignedat} \]
4c (310)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4c}}&{}\approx{}&0.211290286782&{}:{}&1.056730202134&{}:{}&-0.268020488916&,\\ A^\prime_{\mathbf{4c}}&{}\approx{}&4.314755078384&{}:{}&13.259020313537&{}:{}&-16.573775391922&,\\B^\prime_{\mathbf{4c}}&{}\approx{}&0.113130530351&{}:{}&1.075420353567&{}:{}&-0.188550883918&,\\C^\prime_{\mathbf{4c}}&{}\approx{}&0.387704095259&{}:{}&0.516938793679&{}:{}&0.095357111061&,\\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.175152306462&{}:{}&0.949268302909&{}:{}&-0.124420609371&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.535065522666&{}:{}&0.995081636481&{}:{}&-0.530147159147&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.160392468186&{}:{}&1.212467473636&{}:{}&-0.372859941821&, \end{alignedat} \]
4c (310)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4c}}&{}\approx{}&-0.549573490654&{}:{}&1.233738761087&{}:{}&0.315834729567&,\\ A^*_{\mathbf{4c}}&{}\approx{}&0.000000000000&{}:{}&0.796179573623&{}:{}&0.203820426377&,\\B^*_{\mathbf{4c}}&{}\approx{}&2.351229586822&{}:{}&0.000000000000&{}:{}&-1.351229586822&,\\C^*_{\mathbf{4c}}&{}\approx{}&-0.803275925284&{}:{}&1.803275925284&{}:{}&0.000000000000&. \end{alignedat} \]
4c (310)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4c}}&{}\approx{}&0.346053821241&{}:{}&1.098641704032&{}:{}&-0.444695525273&,\\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.175152306462&{}:{}&0.949268302909&{}:{}&-0.124420609371&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.535065522666&{}:{}&0.995081636481&{}:{}&-0.530147159147&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.160392468186&{}:{}&1.212467473636&{}:{}&-0.372859941821&,\\ A^*_{\mathbf{4c}}&{}\approx{}&0.000000000000&{}:{}&0.796179573623&{}:{}&0.203820426377&,\\B^*_{\mathbf{4c}}&{}\approx{}&2.351229586822&{}:{}&0.000000000000&{}:{}&-1.351229586822&,\\C^*_{\mathbf{4c}}&{}\approx{}&-0.803275925284&{}:{}&1.803275925284&{}:{}&0.000000000000&. \end{alignedat} \]
4c (310)