Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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2(020)

Malfatti circles

2 (020)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.250000000000&{}:{}&0.333333333333&{}:{}&0.416666666667&, \\ P_{\mathbf{2}}&{}\approx{}&0.002186055432&{}:{}&0.993066377918&{}:{}&0.004747566651&, \\ P^-_{\mathbf{2}}&{}\approx{}&-0.010110323950&{}:{}&1.025801936124&{}:{}&-0.015691612174&, \\ P^+_{\mathbf{2}}&{}\approx{}&0.013372323027&{}:{}&0.963286171677&{}:{}&0.023341505296&, \\ Q_{\mathbf{2}}&{}\approx{}&-0.044343286406&{}:{}&1.157529479812&{}:{}&-0.113186193407&, \\ I^\prime_{\mathbf{2}}&{}\approx{}&0.036345867083&{}:{}&0.894505378899&{}:{}&0.069148754018&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{2}}\)
\(P^-_{\mathbf{2}}\)
\(P^+_{\mathbf{2}}\)
\(Q_{\mathbf{2}}\)
\(I^\prime_{\mathbf{2}}\)
2 (020)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{2}}&{}\approx{}&-0.373016509412&{}:{}&0.610229559739&{}:{}&0.762786949673&,\\B^\prime_{\mathbf{2}}&{}\approx{}&0.011715049997&{}:{}&0.968759866674&{}:{}&0.019525083329&,\\C^\prime_{\mathbf{2}}&{}\approx{}&0.936000484963&{}:{}&1.248000646617&{}:{}&-1.184001131580&, \\ A^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.023140472166&{}:{}&0.972211661624&{}:{}&0.004647866210&,\\B^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.056855344697&{}:{}&0.819669040503&{}:{}&0.123475614800&,\\C^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.002112851079&{}:{}&0.959811603291&{}:{}&0.038075545630&, \\ A^{\prime\prime\prime}_{\mathbf{2}}&{}\approx{}&0.012131702742&{}:{}&1.003214389493&{}:{}&-0.015346092236&,\\B^{\prime\prime\prime}_{\mathbf{2}}&{}\approx{}&1.091575854718&{}:{}&-1.785743623776&{}:{}&1.694167769059&,\\C^{\prime\prime\prime}_{\mathbf{2}}&{}\approx{}&-0.009755550080&{}:{}&0.989806282152&{}:{}&0.019949267928&, \\ A^*_{\mathbf{2}}&{}\approx{}&0.000000000000&{}:{}&1.108380256645&{}:{}&-0.108380256645&,\\B^*_{\mathbf{2}}&{}\approx{}&0.281491987776&{}:{}&0.000000000000&{}:{}&0.718508012224&,\\C^*_{\mathbf{2}}&{}\approx{}&-0.039834563767&{}:{}&1.039834563767&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{2}}}{B^\prime_{\mathbf{2}}}{C^\prime_{\mathbf{2}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{2}}}{B^{\prime\prime}_{\mathbf{2}}}{C^{\prime\prime}_{\mathbf{2}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{2}}}{B^{\prime\prime\prime}_{\mathbf{2}}}{C^{\prime\prime\prime}_{\mathbf{2}}}\)
\(\triangle{A^*_{\mathbf{2}}}{B^*_{\mathbf{2}}}{C^*_{\mathbf{2}}}\)
2 (020)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{2}}}}&{}\approx{}&1.830688679216&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{2}}}}&{}\approx{}&0.046860199990&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{2}}}}&{}\approx{}&3.744001939852&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.250000000000&{}:{}&0.333333333333&{}:{}&0.416666666667&,\\ A^\prime_{\mathbf{2}}&{}\approx{}&-0.373016509412&{}:{}&0.610229559739&{}:{}&0.762786949673&,\\B^\prime_{\mathbf{2}}&{}\approx{}&0.011715049997&{}:{}&0.968759866674&{}:{}&0.019525083329&,\\C^\prime_{\mathbf{2}}&{}\approx{}&0.936000484963&{}:{}&1.248000646617&{}:{}&-1.184001131580&. \end{alignedat} \]
2 (020)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{2}}&{}\approx{}&0.002186055432&{}:{}&0.993066377918&{}:{}&0.004747566651&,\\ A^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.023140472166&{}:{}&0.972211661624&{}:{}&0.004647866210&,\\B^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.056855344697&{}:{}&0.819669040503&{}:{}&0.123475614800&,\\C^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.002112851079&{}:{}&0.959811603291&{}:{}&0.038075545630&. \end{alignedat} \]
2 (020)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{2}}&{}\approx{}&-0.010110323950&{}:{}&1.025801936124&{}:{}&-0.015691612174&,\\ A^{\prime\prime\prime}_{\mathbf{2}}&{}\approx{}&0.012131702742&{}:{}&1.003214389493&{}:{}&-0.015346092236&,\\B^{\prime\prime\prime}_{\mathbf{2}}&{}\approx{}&1.091575854718&{}:{}&-1.785743623776&{}:{}&1.694167769059&,\\C^{\prime\prime\prime}_{\mathbf{2}}&{}\approx{}&-0.009755550080&{}:{}&0.989806282152&{}:{}&0.019949267928&. \end{alignedat} \]
2 (020)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{2}}&{}\approx{}&0.013372323027&{}:{}&0.963286171677&{}:{}&0.023341505296&,\\ A^\prime_{\mathbf{2}}&{}\approx{}&-0.373016509412&{}:{}&0.610229559739&{}:{}&0.762786949673&,\\B^\prime_{\mathbf{2}}&{}\approx{}&0.011715049997&{}:{}&0.968759866674&{}:{}&0.019525083329&,\\C^\prime_{\mathbf{2}}&{}\approx{}&0.936000484963&{}:{}&1.248000646617&{}:{}&-1.184001131580&,\\ A^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.023140472166&{}:{}&0.972211661624&{}:{}&0.004647866210&,\\B^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.056855344697&{}:{}&0.819669040503&{}:{}&0.123475614800&,\\C^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.002112851079&{}:{}&0.959811603291&{}:{}&0.038075545630&, \end{alignedat} \]
2 (020)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{2}}&{}\approx{}&-0.044343286406&{}:{}&1.157529479812&{}:{}&-0.113186193407&,\\ A^*_{\mathbf{2}}&{}\approx{}&0.000000000000&{}:{}&1.108380256645&{}:{}&-0.108380256645&,\\B^*_{\mathbf{2}}&{}\approx{}&0.281491987776&{}:{}&0.000000000000&{}:{}&0.718508012224&,\\C^*_{\mathbf{2}}&{}\approx{}&-0.039834563767&{}:{}&1.039834563767&{}:{}&0.000000000000&. \end{alignedat} \]
2 (020)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{2}}&{}\approx{}&0.036345867083&{}:{}&0.894505378899&{}:{}&0.069148754018&,\\ A^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.023140472166&{}:{}&0.972211661624&{}:{}&0.004647866210&,\\B^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.056855344697&{}:{}&0.819669040503&{}:{}&0.123475614800&,\\C^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.002112851079&{}:{}&0.959811603291&{}:{}&0.038075545630&,\\ A^*_{\mathbf{2}}&{}\approx{}&0.000000000000&{}:{}&1.108380256645&{}:{}&-0.108380256645&,\\B^*_{\mathbf{2}}&{}\approx{}&0.281491987776&{}:{}&0.000000000000&{}:{}&0.718508012224&,\\C^*_{\mathbf{2}}&{}\approx{}&-0.039834563767&{}:{}&1.039834563767&{}:{}&0.000000000000&. \end{alignedat} \]
2 (020)