Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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Malfatti circles

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Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.250000000000&{}:{}&0.333333333333&{}:{}&0.416666666667&, \\ P&{}\approx{}&0.218219101089&{}:{}&0.307863489535&{}:{}&0.473917409375&, \\ P^-&{}\approx{}&0.206879095206&{}:{}&0.298775383522&{}:{}&0.494345521272&, \\ P^+&{}\approx{}&0.224836610348&{}:{}&0.313166892929&{}:{}&0.461996496723&, \\ Q&{}\approx{}&0.199711799657&{}:{}&0.290524035034&{}:{}&0.509764165309&, \\ I^\prime&{}\approx{}&0.233970657965&{}:{}&0.320895389436&{}:{}&0.445133952599&, \end{alignedat} \]
\(I\)
\(P\)
\(P^-\)
\(P^+\)
\(Q\)
\(I^\prime\)
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Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.436000484963&{}:{}&0.250666451128&{}:{}&0.313333063909&,\\B^\prime&{}\approx{}&0.166223547185&{}:{}&0.556737207507&{}:{}&0.277039245308&,\\C^\prime&{}\approx{}&0.126983490588&{}:{}&0.169311320784&{}:{}&0.703705188628&, \\ A^{\prime\prime}&{}\approx{}&0.143977759878&{}:{}&0.337099556066&{}:{}&0.518922684056&,\\B^{\prime\prime}&{}\approx{}&0.251561692011&{}:{}&0.202109115078&{}:{}&0.546329192911&,\\C^{\prime\prime}&{}\approx{}&0.292819591879&{}:{}&0.413109855691&{}:{}&0.294070552430&, \\ A^{\prime\prime\prime}&{}\approx{}&0.101101661394&{}:{}&0.338622641568&{}:{}&0.560275697038&,\\B^{\prime\prime\prime}&{}\approx{}&0.252241117390&{}:{}&0.145019072008&{}:{}&0.602739810602&,\\C^{\prime\prime\prime}&{}\approx{}&0.316169953677&{}:{}&0.456613555245&{}:{}&0.227216491078&, \\ A^*&{}\approx{}&0.000000000000&{}:{}&0.363024264146&{}:{}&0.636975735854&,\\B^*&{}\approx{}&0.281491987776&{}:{}&0.000000000000&{}:{}&0.718508012224&,\\C^*&{}\approx{}&0.407379031733&{}:{}&0.592620968267&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime}{B^\prime}{C^\prime}\)
\(\triangle{A^{\prime\prime}}{B^{\prime\prime}}{C^{\prime\prime}}\)
\(\triangle{A^{\prime\prime\prime}}{B^{\prime\prime\prime}}{C^{\prime\prime\prime}}\)
\(\triangle{A^*}{B^*}{C^*}\)
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Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime}}&{}\approx{}&0.751999353383&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime}}&{}\approx{}&0.664894188740&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime}}&{}\approx{}&0.507933962352&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.250000000000&{}:{}&0.333333333333&{}:{}&0.416666666667&,\\ A^\prime&{}\approx{}&0.436000484963&{}:{}&0.250666451128&{}:{}&0.313333063909&,\\B^\prime&{}\approx{}&0.166223547185&{}:{}&0.556737207507&{}:{}&0.277039245308&,\\C^\prime&{}\approx{}&0.126983490588&{}:{}&0.169311320784&{}:{}&0.703705188628&. \end{alignedat} \]
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First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P&{}\approx{}&0.218219101089&{}:{}&0.307863489535&{}:{}&0.473917409375&,\\ A^{\prime\prime}&{}\approx{}&0.143977759878&{}:{}&0.337099556066&{}:{}&0.518922684056&,\\B^{\prime\prime}&{}\approx{}&0.251561692011&{}:{}&0.202109115078&{}:{}&0.546329192911&,\\C^{\prime\prime}&{}\approx{}&0.292819591879&{}:{}&0.413109855691&{}:{}&0.294070552430&. \end{alignedat} \]
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First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-&{}\approx{}&0.206879095206&{}:{}&0.298775383522&{}:{}&0.494345521272&,\\ A^{\prime\prime\prime}&{}\approx{}&0.101101661394&{}:{}&0.338622641568&{}:{}&0.560275697038&,\\B^{\prime\prime\prime}&{}\approx{}&0.252241117390&{}:{}&0.145019072008&{}:{}&0.602739810602&,\\C^{\prime\prime\prime}&{}\approx{}&0.316169953677&{}:{}&0.456613555245&{}:{}&0.227216491078&. \end{alignedat} \]
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Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+&{}\approx{}&0.224836610348&{}:{}&0.313166892929&{}:{}&0.461996496723&,\\ A^\prime&{}\approx{}&0.436000484963&{}:{}&0.250666451128&{}:{}&0.313333063909&,\\B^\prime&{}\approx{}&0.166223547185&{}:{}&0.556737207507&{}:{}&0.277039245308&,\\C^\prime&{}\approx{}&0.126983490588&{}:{}&0.169311320784&{}:{}&0.703705188628&,\\ A^{\prime\prime}&{}\approx{}&0.143977759878&{}:{}&0.337099556066&{}:{}&0.518922684056&,\\B^{\prime\prime}&{}\approx{}&0.251561692011&{}:{}&0.202109115078&{}:{}&0.546329192911&,\\C^{\prime\prime}&{}\approx{}&0.292819591879&{}:{}&0.413109855691&{}:{}&0.294070552430&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q&{}\approx{}&0.199711799657&{}:{}&0.290524035034&{}:{}&0.509764165309&,\\ A^*&{}\approx{}&0.000000000000&{}:{}&0.363024264146&{}:{}&0.636975735854&,\\B^*&{}\approx{}&0.281491987776&{}:{}&0.000000000000&{}:{}&0.718508012224&,\\C^*&{}\approx{}&0.407379031733&{}:{}&0.592620968267&{}:{}&0.000000000000&. \end{alignedat} \]
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Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.233970657965&{}:{}&0.320895389436&{}:{}&0.445133952599&,\\ A^{\prime\prime}&{}\approx{}&0.143977759878&{}:{}&0.337099556066&{}:{}&0.518922684056&,\\B^{\prime\prime}&{}\approx{}&0.251561692011&{}:{}&0.202109115078&{}:{}&0.546329192911&,\\C^{\prime\prime}&{}\approx{}&0.292819591879&{}:{}&0.413109855691&{}:{}&0.294070552430&,\\ A^*&{}\approx{}&0.000000000000&{}:{}&0.363024264146&{}:{}&0.636975735854&,\\B^*&{}\approx{}&0.281491987776&{}:{}&0.000000000000&{}:{}&0.718508012224&,\\C^*&{}\approx{}&0.407379031733&{}:{}&0.592620968267&{}:{}&0.000000000000&. \end{alignedat} \]
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