Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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0a(011)

Malfatti circles

0a (011)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.500000000000&{}:{}&0.666666666667&{}:{}&0.833333333333&, \\ P_{\mathbf{a}}&{}\approx{}&-0.278614241899&{}:{}&0.673533664275&{}:{}&0.605080577624&, \\ P^-_{\mathbf{a}}&{}\approx{}&-0.216258929939&{}:{}&0.675467816597&{}:{}&0.540791113342&, \\ P^+_{\mathbf{a}}&{}\approx{}&-0.318500755387&{}:{}&0.672296454641&{}:{}&0.646204300746&, \\ Q_{\mathbf{a}}&{}\approx{}&-0.186532501706&{}:{}&0.710408481001&{}:{}&0.476124020704&, \\ I^\prime_{\mathbf{a}}&{}\approx{}&-0.370663725296&{}:{}&0.665468421915&{}:{}&0.705195303382&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{a}}\)
\(P^-_{\mathbf{a}}\)
\(P^+_{\mathbf{a}}\)
\(Q_{\mathbf{a}}\)
\(I^\prime_{\mathbf{a}}\)
0a (011)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{a}}&{}\approx{}&-1.048631302823&{}:{}&0.910502801255&{}:{}&1.138128501568&,\\B^\prime_{\mathbf{a}}&{}\approx{}&-0.183049043273&{}:{}&0.877967304485&{}:{}&0.305081738788&,\\C^\prime_{\mathbf{a}}&{}\approx{}&-0.239614308448&{}:{}&0.319485744597&{}:{}&0.920128563851&, \\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.207546595830&{}:{}&0.636097469292&{}:{}&0.571449126538&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.449753986343&{}:{}&0.473000608660&{}:{}&0.976753377682&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.366025392403&{}:{}&0.884845017551&{}:{}&0.481180374852&, \\ A^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.130951923948&{}:{}&0.628091278872&{}:{}&0.502860645075&,\\B^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.422129330949&{}:{}&0.366525333820&{}:{}&1.055603997129&,\\C^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.311646061027&{}:{}&0.973402043800&{}:{}&0.338244017226&, \\ A^*_{\mathbf{a}}&{}\approx{}&0.000000000000&{}:{}&0.598726524541&{}:{}&0.401273475459&,\\B^*_{\mathbf{a}}&{}\approx{}&-0.644122805635&{}:{}&0.000000000000&{}:{}&1.644122805635&,\\C^*_{\mathbf{a}}&{}\approx{}&-0.356062329784&{}:{}&1.356062329784&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{a}}}{B^\prime_{\mathbf{a}}}{C^\prime_{\mathbf{a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{a}}}{B^{\prime\prime}_{\mathbf{a}}}{C^{\prime\prime}_{\mathbf{a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{a}}}{B^{\prime\prime\prime}_{\mathbf{a}}}{C^{\prime\prime\prime}_{\mathbf{a}}}\)
\(\triangle{A^*_{\mathbf{a}}}{B^*_{\mathbf{a}}}{C^*_{\mathbf{a}}}\)
0a (011)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{a}}}}&{}\approx{}&1.365754201882&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{a}}}}&{}\approx{}&0.366098086545&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{a}}}}&{}\approx{}&0.479228616896&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.500000000000&{}:{}&0.666666666667&{}:{}&0.833333333333&,\\ A^\prime_{\mathbf{a}}&{}\approx{}&-1.048631302823&{}:{}&0.910502801255&{}:{}&1.138128501568&,\\B^\prime_{\mathbf{a}}&{}\approx{}&-0.183049043273&{}:{}&0.877967304485&{}:{}&0.305081738788&,\\C^\prime_{\mathbf{a}}&{}\approx{}&-0.239614308448&{}:{}&0.319485744597&{}:{}&0.920128563851&. \end{alignedat} \]
0a (011)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{a}}&{}\approx{}&-0.278614241899&{}:{}&0.673533664275&{}:{}&0.605080577624&,\\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.207546595830&{}:{}&0.636097469292&{}:{}&0.571449126538&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.449753986343&{}:{}&0.473000608660&{}:{}&0.976753377682&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.366025392403&{}:{}&0.884845017551&{}:{}&0.481180374852&. \end{alignedat} \]
0a (011)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{a}}&{}\approx{}&-0.216258929939&{}:{}&0.675467816597&{}:{}&0.540791113342&,\\ A^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.130951923948&{}:{}&0.628091278872&{}:{}&0.502860645075&,\\B^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.422129330949&{}:{}&0.366525333820&{}:{}&1.055603997129&,\\C^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.311646061027&{}:{}&0.973402043800&{}:{}&0.338244017226&. \end{alignedat} \]
0a (011)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{a}}&{}\approx{}&-0.318500755387&{}:{}&0.672296454641&{}:{}&0.646204300746&,\\ A^\prime_{\mathbf{a}}&{}\approx{}&-1.048631302823&{}:{}&0.910502801255&{}:{}&1.138128501568&,\\B^\prime_{\mathbf{a}}&{}\approx{}&-0.183049043273&{}:{}&0.877967304485&{}:{}&0.305081738788&,\\C^\prime_{\mathbf{a}}&{}\approx{}&-0.239614308448&{}:{}&0.319485744597&{}:{}&0.920128563851&,\\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.207546595830&{}:{}&0.636097469292&{}:{}&0.571449126538&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.449753986343&{}:{}&0.473000608660&{}:{}&0.976753377682&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.366025392403&{}:{}&0.884845017551&{}:{}&0.481180374852&, \end{alignedat} \]
0a (011)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{a}}&{}\approx{}&-0.186532501706&{}:{}&0.710408481001&{}:{}&0.476124020704&,\\ A^*_{\mathbf{a}}&{}\approx{}&0.000000000000&{}:{}&0.598726524541&{}:{}&0.401273475459&,\\B^*_{\mathbf{a}}&{}\approx{}&-0.644122805635&{}:{}&0.000000000000&{}:{}&1.644122805635&,\\C^*_{\mathbf{a}}&{}\approx{}&-0.356062329784&{}:{}&1.356062329784&{}:{}&0.000000000000&. \end{alignedat} \]
0a (011)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{a}}&{}\approx{}&-0.370663725296&{}:{}&0.665468421915&{}:{}&0.705195303382&,\\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.207546595830&{}:{}&0.636097469292&{}:{}&0.571449126538&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.449753986343&{}:{}&0.473000608660&{}:{}&0.976753377682&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.366025392403&{}:{}&0.884845017551&{}:{}&0.481180374852&,\\ A^*_{\mathbf{a}}&{}\approx{}&0.000000000000&{}:{}&0.598726524541&{}:{}&0.401273475459&,\\B^*_{\mathbf{a}}&{}\approx{}&-0.644122805635&{}:{}&0.000000000000&{}:{}&1.644122805635&,\\C^*_{\mathbf{a}}&{}\approx{}&-0.356062329784&{}:{}&1.356062329784&{}:{}&0.000000000000&. \end{alignedat} \]
0a (011)