Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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3(022)

Malfatti circles

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Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.250000000000&{}:{}&0.333333333333&{}:{}&0.416666666667&, \\ P_{\mathbf{3}}&{}\approx{}&0.001890184895&{}:{}&0.858660325171&{}:{}&0.139449489934&, \\ P^-_{\mathbf{3}}&{}\approx{}&-0.075705892763&{}:{}&1.022955776511&{}:{}&0.052750116252&, \\ P^+_{\mathbf{3}}&{}\approx{}&0.049626992282&{}:{}&0.757586397830&{}:{}&0.192786609888&, \\ Q_{\mathbf{3}}&{}\approx{}&-0.025011954860&{}:{}&0.652907744206&{}:{}&0.372104210653&, \\ I^\prime_{\mathbf{3}}&{}\approx{}&0.027248234267&{}:{}&0.670604227497&{}:{}&0.302147538235&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{3}}\)
\(P^-_{\mathbf{3}}\)
\(P^+_{\mathbf{3}}\)
\(Q_{\mathbf{3}}\)
\(I^\prime_{\mathbf{3}}\)
3 (022)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3}}&{}\approx{}&-0.476569900005&{}:{}&0.656253288891&{}:{}&0.820316611114&,\\B^\prime_{\mathbf{3}}&{}\approx{}&0.063491745294&{}:{}&0.830688679216&{}:{}&0.105819575490&,\\C^\prime_{\mathbf{3}}&{}\approx{}&0.332447094370&{}:{}&0.443262792493&{}:{}&0.224290113137&, \\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.106620741008&{}:{}&0.768562049403&{}:{}&0.124817209588&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.006297128787&{}:{}&0.529128532709&{}:{}&0.464574338503&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.001784736249&{}:{}&0.810757831872&{}:{}&0.187457431879&, \\ A^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&0.067759508522&{}:{}&0.886525584986&{}:{}&0.045714906492&,\\B^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&-0.931262069901&{}:{}&1.282380184282&{}:{}&0.648881885619&,\\C^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&-0.070256650945&{}:{}&0.949324343195&{}:{}&0.120932307750&, \\ A^*_{\mathbf{3}}&{}\approx{}&0.000000000000&{}:{}&0.636975735854&{}:{}&0.363024264146&,\\B^*_{\mathbf{3}}&{}\approx{}&-0.072061402818&{}:{}&0.000000000000&{}:{}&1.072061402818&,\\C^*_{\mathbf{3}}&{}\approx{}&-0.039834563767&{}:{}&1.039834563767&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3}}}{B^\prime_{\mathbf{3}}}{C^\prime_{\mathbf{3}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3}}}{B^{\prime\prime}_{\mathbf{3}}}{C^{\prime\prime}_{\mathbf{3}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3}}}{B^{\prime\prime\prime}_{\mathbf{3}}}{C^{\prime\prime\prime}_{\mathbf{3}}}\)
\(\triangle{A^*_{\mathbf{3}}}{B^*_{\mathbf{3}}}{C^*_{\mathbf{3}}}\)
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Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3}}}}&{}\approx{}&1.968759866674&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3}}}}&{}\approx{}&0.253966981176&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3}}}}&{}\approx{}&1.329788377479&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.250000000000&{}:{}&0.333333333333&{}:{}&0.416666666667&,\\ A^\prime_{\mathbf{3}}&{}\approx{}&-0.476569900005&{}:{}&0.656253288891&{}:{}&0.820316611114&,\\B^\prime_{\mathbf{3}}&{}\approx{}&0.063491745294&{}:{}&0.830688679216&{}:{}&0.105819575490&,\\C^\prime_{\mathbf{3}}&{}\approx{}&0.332447094370&{}:{}&0.443262792493&{}:{}&0.224290113137&. \end{alignedat} \]
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First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3}}&{}\approx{}&0.001890184895&{}:{}&0.858660325171&{}:{}&0.139449489934&,\\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.106620741008&{}:{}&0.768562049403&{}:{}&0.124817209588&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.006297128787&{}:{}&0.529128532709&{}:{}&0.464574338503&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.001784736249&{}:{}&0.810757831872&{}:{}&0.187457431879&. \end{alignedat} \]
3 (022)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3}}&{}\approx{}&-0.075705892763&{}:{}&1.022955776511&{}:{}&0.052750116252&,\\ A^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&0.067759508522&{}:{}&0.886525584986&{}:{}&0.045714906492&,\\B^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&-0.931262069901&{}:{}&1.282380184282&{}:{}&0.648881885619&,\\C^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&-0.070256650945&{}:{}&0.949324343195&{}:{}&0.120932307750&. \end{alignedat} \]
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Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3}}&{}\approx{}&0.049626992282&{}:{}&0.757586397830&{}:{}&0.192786609888&,\\ A^\prime_{\mathbf{3}}&{}\approx{}&-0.476569900005&{}:{}&0.656253288891&{}:{}&0.820316611114&,\\B^\prime_{\mathbf{3}}&{}\approx{}&0.063491745294&{}:{}&0.830688679216&{}:{}&0.105819575490&,\\C^\prime_{\mathbf{3}}&{}\approx{}&0.332447094370&{}:{}&0.443262792493&{}:{}&0.224290113137&,\\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.106620741008&{}:{}&0.768562049403&{}:{}&0.124817209588&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.006297128787&{}:{}&0.529128532709&{}:{}&0.464574338503&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.001784736249&{}:{}&0.810757831872&{}:{}&0.187457431879&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3}}&{}\approx{}&-0.025011954860&{}:{}&0.652907744206&{}:{}&0.372104210653&,\\ A^*_{\mathbf{3}}&{}\approx{}&0.000000000000&{}:{}&0.636975735854&{}:{}&0.363024264146&,\\B^*_{\mathbf{3}}&{}\approx{}&-0.072061402818&{}:{}&0.000000000000&{}:{}&1.072061402818&,\\C^*_{\mathbf{3}}&{}\approx{}&-0.039834563767&{}:{}&1.039834563767&{}:{}&0.000000000000&. \end{alignedat} \]
3 (022)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3}}&{}\approx{}&0.027248234267&{}:{}&0.670604227497&{}:{}&0.302147538235&,\\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.106620741008&{}:{}&0.768562049403&{}:{}&0.124817209588&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.006297128787&{}:{}&0.529128532709&{}:{}&0.464574338503&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.001784736249&{}:{}&0.810757831872&{}:{}&0.187457431879&,\\ A^*_{\mathbf{3}}&{}\approx{}&0.000000000000&{}:{}&0.636975735854&{}:{}&0.363024264146&,\\B^*_{\mathbf{3}}&{}\approx{}&-0.072061402818&{}:{}&0.000000000000&{}:{}&1.072061402818&,\\C^*_{\mathbf{3}}&{}\approx{}&-0.039834563767&{}:{}&1.039834563767&{}:{}&0.000000000000&. \end{alignedat} \]
3 (022)