Derousseau's Generalization of the Malfatti circles

Pythagorean Triangle U

\(C=90\degree\).   \(a:b:c=5:12:13\).


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2(020)

Malfatti circles

2 (020)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.166666666667&{}:{}&0.400000000000&{}:{}&0.433333333333&, \\ P_{\mathbf{2}}&{}\approx{}&0.002956284236&{}:{}&0.986697459400&{}:{}&0.010346256364&, \\ P^-_{\mathbf{2}}&{}\approx{}&-0.007728204711&{}:{}&1.024988021230&{}:{}&-0.017259816519&, \\ P^+_{\mathbf{2}}&{}\approx{}&0.012407158546&{}:{}&0.952827866866&{}:{}&0.034764974588&, \\ Q_{\mathbf{2}}&{}\approx{}&-0.035496684994&{}:{}&1.183984672971&{}:{}&-0.148487987977&, \\ I^\prime_{\mathbf{2}}&{}\approx{}&0.030941555648&{}:{}&0.875722840244&{}:{}&0.093335604108&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{2}}\)
\(P^-_{\mathbf{2}}\)
\(P^+_{\mathbf{2}}\)
\(Q_{\mathbf{2}}\)
\(I^\prime_{\mathbf{2}}\)
2 (020)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{2}}&{}\approx{}&-0.165631799079&{}:{}&0.559503263558&{}:{}&0.606128535521&,\\B^\prime_{\mathbf{2}}&{}\approx{}&0.010221738128&{}:{}&0.963201742740&{}:{}&0.026576519133&,\\C^\prime_{\mathbf{2}}&{}\approx{}&0.551497473876&{}:{}&1.323593937302&{}:{}&-0.875091411178&, \\ A^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.020071461455&{}:{}&0.969759885238&{}:{}&0.010168653307&,\\B^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.047440669208&{}:{}&0.786528838987&{}:{}&0.166030491805&,\\C^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.002835081641&{}:{}&0.946244551902&{}:{}&0.050920366457&, \\ A^{\prime\prime\prime}_{\mathbf{2}}&{}\approx{}&0.010678744770&{}:{}&1.006265807604&{}:{}&-0.016944552374&,\\B^{\prime\prime\prime}_{\mathbf{2}}&{}\approx{}&-6.913702863200&{}:{}&23.354448461788&{}:{}&-15.440745598588&,\\C^{\prime\prime\prime}_{\mathbf{2}}&{}\approx{}&-0.007391583616&{}:{}&0.980342129483&{}:{}&0.027049454133&, \\ A^*_{\mathbf{2}}&{}\approx{}&0.000000000000&{}:{}&1.143397840021&{}:{}&-0.143397840021&,\\B^*_{\mathbf{2}}&{}\approx{}&0.192932837398&{}:{}&0.000000000000&{}:{}&0.807067162602&,\\C^*_{\mathbf{2}}&{}\approx{}&-0.030907319333&{}:{}&1.030907319333&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{2}}}{B^\prime_{\mathbf{2}}}{C^\prime_{\mathbf{2}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{2}}}{B^{\prime\prime}_{\mathbf{2}}}{C^{\prime\prime}_{\mathbf{2}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{2}}}{B^{\prime\prime\prime}_{\mathbf{2}}}{C^{\prime\prime\prime}_{\mathbf{2}}}\)
\(\triangle{A^*_{\mathbf{2}}}{B^*_{\mathbf{2}}}{C^*_{\mathbf{2}}}\)
2 (020)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{2}}}}&{}\approx{}&1.398758158895&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{2}}}}&{}\approx{}&0.061330428767&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{2}}}}&{}\approx{}&3.308984843256&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.166666666667&{}:{}&0.400000000000&{}:{}&0.433333333333&,\\ A^\prime_{\mathbf{2}}&{}\approx{}&-0.165631799079&{}:{}&0.559503263558&{}:{}&0.606128535521&,\\B^\prime_{\mathbf{2}}&{}\approx{}&0.010221738128&{}:{}&0.963201742740&{}:{}&0.026576519133&,\\C^\prime_{\mathbf{2}}&{}\approx{}&0.551497473876&{}:{}&1.323593937302&{}:{}&-0.875091411178&. \end{alignedat} \]
2 (020)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{2}}&{}\approx{}&0.002956284236&{}:{}&0.986697459400&{}:{}&0.010346256364&,\\ A^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.020071461455&{}:{}&0.969759885238&{}:{}&0.010168653307&,\\B^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.047440669208&{}:{}&0.786528838987&{}:{}&0.166030491805&,\\C^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.002835081641&{}:{}&0.946244551902&{}:{}&0.050920366457&. \end{alignedat} \]
2 (020)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{2}}&{}\approx{}&-0.007728204711&{}:{}&1.024988021230&{}:{}&-0.017259816519&,\\ A^{\prime\prime\prime}_{\mathbf{2}}&{}\approx{}&0.010678744770&{}:{}&1.006265807604&{}:{}&-0.016944552374&,\\B^{\prime\prime\prime}_{\mathbf{2}}&{}\approx{}&-6.913702863200&{}:{}&23.354448461788&{}:{}&-15.440745598588&,\\C^{\prime\prime\prime}_{\mathbf{2}}&{}\approx{}&-0.007391583616&{}:{}&0.980342129483&{}:{}&0.027049454133&. \end{alignedat} \]
2 (020)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{2}}&{}\approx{}&0.012407158546&{}:{}&0.952827866866&{}:{}&0.034764974588&,\\ A^\prime_{\mathbf{2}}&{}\approx{}&-0.165631799079&{}:{}&0.559503263558&{}:{}&0.606128535521&,\\B^\prime_{\mathbf{2}}&{}\approx{}&0.010221738128&{}:{}&0.963201742740&{}:{}&0.026576519133&,\\C^\prime_{\mathbf{2}}&{}\approx{}&0.551497473876&{}:{}&1.323593937302&{}:{}&-0.875091411178&,\\ A^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.020071461455&{}:{}&0.969759885238&{}:{}&0.010168653307&,\\B^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.047440669208&{}:{}&0.786528838987&{}:{}&0.166030491805&,\\C^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.002835081641&{}:{}&0.946244551902&{}:{}&0.050920366457&, \end{alignedat} \]
2 (020)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{2}}&{}\approx{}&-0.035496684994&{}:{}&1.183984672971&{}:{}&-0.148487987977&,\\ A^*_{\mathbf{2}}&{}\approx{}&0.000000000000&{}:{}&1.143397840021&{}:{}&-0.143397840021&,\\B^*_{\mathbf{2}}&{}\approx{}&0.192932837398&{}:{}&0.000000000000&{}:{}&0.807067162602&,\\C^*_{\mathbf{2}}&{}\approx{}&-0.030907319333&{}:{}&1.030907319333&{}:{}&0.000000000000&. \end{alignedat} \]
2 (020)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{2}}&{}\approx{}&0.030941555648&{}:{}&0.875722840244&{}:{}&0.093335604108&,\\ A^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.020071461455&{}:{}&0.969759885238&{}:{}&0.010168653307&,\\B^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.047440669208&{}:{}&0.786528838987&{}:{}&0.166030491805&,\\C^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.002835081641&{}:{}&0.946244551902&{}:{}&0.050920366457&,\\ A^*_{\mathbf{2}}&{}\approx{}&0.000000000000&{}:{}&1.143397840021&{}:{}&-0.143397840021&,\\B^*_{\mathbf{2}}&{}\approx{}&0.192932837398&{}:{}&0.000000000000&{}:{}&0.807067162602&,\\C^*_{\mathbf{2}}&{}\approx{}&-0.030907319333&{}:{}&1.030907319333&{}:{}&0.000000000000&. \end{alignedat} \]
2 (020)