Derousseau's Generalization of the Malfatti circles

Pythagorean Triangle U

\(C=90\degree\).   \(a:b:c=5:12:13\).


[Top] > Pythagorean Triangle U > 0 (000)

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Malfatti circles

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Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.166666666667&{}:{}&0.400000000000&{}:{}&0.433333333333&, \\ P&{}\approx{}&0.136898592336&{}:{}&0.383990531012&{}:{}&0.479110876652&, \\ P^-&{}\approx{}&0.126434937858&{}:{}&0.378363107821&{}:{}&0.495201954321&, \\ P^+&{}\approx{}&0.143042797667&{}:{}&0.387294925634&{}:{}&0.469662276699&, \\ Q&{}\approx{}&0.121346147327&{}:{}&0.371044613433&{}:{}&0.507609239240&, \\ I^\prime&{}\approx{}&0.151261011934&{}:{}&0.392458182810&{}:{}&0.456280805256&, \end{alignedat} \]
\(I\)
\(P\)
\(P^-\)
\(P^+\)
\(Q\)
\(I^\prime\)
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Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.301497473876&{}:{}&0.335281212540&{}:{}&0.363221313584&,\\B^\prime&{}\approx{}&0.099419031209&{}:{}&0.642091487647&{}:{}&0.258489481144&,\\C^\prime&{}\approx{}&0.084368200921&{}:{}&0.202483682210&{}:{}&0.713148116870&, \\ A^{\prime\prime}&{}\approx{}&0.091277339597&{}:{}&0.404287252706&{}:{}&0.504435407697&,\\B^{\prime\prime}&{}\approx{}&0.166123509110&{}:{}&0.252485705756&{}:{}&0.581390785134&,\\C^{\prime\prime}&{}\approx{}&0.183437440152&{}:{}&0.514528592657&{}:{}&0.302033967192&, \\ A^{\prime\prime\prime}&{}\approx{}&0.062848639681&{}:{}&0.405903940709&{}:{}&0.531247419610&,\\B^{\prime\prime\prime}&{}\approx{}&0.165873032509&{}:{}&0.184459626651&{}:{}&0.649667340840&,\\C^{\prime\prime\prime}&{}\approx{}&0.192408481420&{}:{}&0.575792357984&{}:{}&0.231799160596&, \\ A^*&{}\approx{}&0.000000000000&{}:{}&0.422287584928&{}:{}&0.577712415072&,\\B^*&{}\approx{}&0.192932837398&{}:{}&0.000000000000&{}:{}&0.807067162602&,\\C^*&{}\approx{}&0.246442778780&{}:{}&0.753557221220&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime}{B^\prime}{C^\prime}\)
\(\triangle{A^{\prime\prime}}{B^{\prime\prime}}{C^{\prime\prime}}\)
\(\triangle{A^{\prime\prime\prime}}{B^{\prime\prime\prime}}{C^{\prime\prime\prime}}\)
\(\triangle{A^*}{B^*}{C^*}\)
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Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime}}&{}\approx{}&0.838203031349&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime}}&{}\approx{}&0.596514187255&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime}}&{}\approx{}&0.506209205524&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.166666666667&{}:{}&0.400000000000&{}:{}&0.433333333333&,\\ A^\prime&{}\approx{}&0.301497473876&{}:{}&0.335281212540&{}:{}&0.363221313584&,\\B^\prime&{}\approx{}&0.099419031209&{}:{}&0.642091487647&{}:{}&0.258489481144&,\\C^\prime&{}\approx{}&0.084368200921&{}:{}&0.202483682210&{}:{}&0.713148116870&. \end{alignedat} \]
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First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P&{}\approx{}&0.136898592336&{}:{}&0.383990531012&{}:{}&0.479110876652&,\\ A^{\prime\prime}&{}\approx{}&0.091277339597&{}:{}&0.404287252706&{}:{}&0.504435407697&,\\B^{\prime\prime}&{}\approx{}&0.166123509110&{}:{}&0.252485705756&{}:{}&0.581390785134&,\\C^{\prime\prime}&{}\approx{}&0.183437440152&{}:{}&0.514528592657&{}:{}&0.302033967192&. \end{alignedat} \]
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First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-&{}\approx{}&0.126434937858&{}:{}&0.378363107821&{}:{}&0.495201954321&,\\ A^{\prime\prime\prime}&{}\approx{}&0.062848639681&{}:{}&0.405903940709&{}:{}&0.531247419610&,\\B^{\prime\prime\prime}&{}\approx{}&0.165873032509&{}:{}&0.184459626651&{}:{}&0.649667340840&,\\C^{\prime\prime\prime}&{}\approx{}&0.192408481420&{}:{}&0.575792357984&{}:{}&0.231799160596&. \end{alignedat} \]
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Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+&{}\approx{}&0.143042797667&{}:{}&0.387294925634&{}:{}&0.469662276699&,\\ A^\prime&{}\approx{}&0.301497473876&{}:{}&0.335281212540&{}:{}&0.363221313584&,\\B^\prime&{}\approx{}&0.099419031209&{}:{}&0.642091487647&{}:{}&0.258489481144&,\\C^\prime&{}\approx{}&0.084368200921&{}:{}&0.202483682210&{}:{}&0.713148116870&,\\ A^{\prime\prime}&{}\approx{}&0.091277339597&{}:{}&0.404287252706&{}:{}&0.504435407697&,\\B^{\prime\prime}&{}\approx{}&0.166123509110&{}:{}&0.252485705756&{}:{}&0.581390785134&,\\C^{\prime\prime}&{}\approx{}&0.183437440152&{}:{}&0.514528592657&{}:{}&0.302033967192&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q&{}\approx{}&0.121346147327&{}:{}&0.371044613433&{}:{}&0.507609239240&,\\ A^*&{}\approx{}&0.000000000000&{}:{}&0.422287584928&{}:{}&0.577712415072&,\\B^*&{}\approx{}&0.192932837398&{}:{}&0.000000000000&{}:{}&0.807067162602&,\\C^*&{}\approx{}&0.246442778780&{}:{}&0.753557221220&{}:{}&0.000000000000&. \end{alignedat} \]
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Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.151261011934&{}:{}&0.392458182810&{}:{}&0.456280805256&,\\ A^{\prime\prime}&{}\approx{}&0.091277339597&{}:{}&0.404287252706&{}:{}&0.504435407697&,\\B^{\prime\prime}&{}\approx{}&0.166123509110&{}:{}&0.252485705756&{}:{}&0.581390785134&,\\C^{\prime\prime}&{}\approx{}&0.183437440152&{}:{}&0.514528592657&{}:{}&0.302033967192&,\\ A^*&{}\approx{}&0.000000000000&{}:{}&0.422287584928&{}:{}&0.577712415072&,\\B^*&{}\approx{}&0.192932837398&{}:{}&0.000000000000&{}:{}&0.807067162602&,\\C^*&{}\approx{}&0.246442778780&{}:{}&0.753557221220&{}:{}&0.000000000000&. \end{alignedat} \]
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