Derousseau's Generalization of the Malfatti circles

Test Case in ETC

The test case used in “Encyclopedis of Triangle Centers&rdquot; to search for triangle centers.

\(a:b:c=6:9:13\).


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Malfatti circles

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Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.214285714286&{}:{}&0.321428571429&{}:{}&0.464285714286&, \\ P&{}\approx{}&0.159447590766&{}:{}&0.246863875275&{}:{}&0.593688533959&, \\ P^-&{}\approx{}&0.141123855203&{}:{}&0.221948657777&{}:{}&0.636927487020&, \\ P^+&{}\approx{}&0.170431169963&{}:{}&0.261798508208&{}:{}&0.567770321829&, \\ Q&{}\approx{}&0.124506936676&{}:{}&0.195962661716&{}:{}&0.679530401609&, \\ I^\prime&{}\approx{}&0.186419499351&{}:{}&0.284090530142&{}:{}&0.529489970507&, \end{alignedat} \]
\(I\)
\(P\)
\(P^-\)
\(P^+\)
\(Q\)
\(I^\prime\)
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Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.349607047620&{}:{}&0.266069844155&{}:{}&0.384323108224&,\\B^\prime&{}\approx{}&0.159580615309&{}:{}&0.494661384854&{}:{}&0.345757999837&,\\C^\prime&{}\approx{}&0.087754730048&{}:{}&0.131632095073&{}:{}&0.780613174879&, \\ A^{\prime\prime}&{}\approx{}&0.113238597558&{}:{}&0.260435106540&{}:{}&0.626326295902&,\\B^{\prime\prime}&{}\approx{}&0.174423075766&{}:{}&0.176128540332&{}:{}&0.649448383901&,\\C^{\prime\prime}&{}\approx{}&0.249574943921&{}:{}&0.386403065307&{}:{}&0.364021990772&, \\ A^{\prime\prime\prime}&{}\approx{}&0.076951751851&{}:{}&0.238531854658&{}:{}&0.684516393491&,\\B^{\prime\prime\prime}&{}\approx{}&0.159380182962&{}:{}&0.121296926738&{}:{}&0.719322890300&,\\C^{\prime\prime\prime}&{}\approx{}&0.272329471274&{}:{}&0.428298678033&{}:{}&0.299371850692&, \\ A^*&{}\approx{}&0.000000000000&{}:{}&0.223831198584&{}:{}&0.776168801416&,\\B^*&{}\approx{}&0.154852182538&{}:{}&0.000000000000&{}:{}&0.845147817462&,\\C^*&{}\approx{}&0.388514034719&{}:{}&0.611485965281&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime}{B^\prime}{C^\prime}\)
\(\triangle{A^{\prime\prime}}{B^{\prime\prime}}{C^{\prime\prime}}\)
\(\triangle{A^{\prime\prime\prime}}{B^{\prime\prime\prime}}{C^{\prime\prime\prime}}\)
\(\triangle{A^*}{B^*}{C^*}\)
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Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime}}&{}\approx{}&0.827772848483&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime}}&{}\approx{}&0.744709538111&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime}}&{}\approx{}&0.409522073559&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.214285714286&{}:{}&0.321428571429&{}:{}&0.464285714286&,\\ A^\prime&{}\approx{}&0.349607047620&{}:{}&0.266069844155&{}:{}&0.384323108224&,\\B^\prime&{}\approx{}&0.159580615309&{}:{}&0.494661384854&{}:{}&0.345757999837&,\\C^\prime&{}\approx{}&0.087754730048&{}:{}&0.131632095073&{}:{}&0.780613174879&. \end{alignedat} \]
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First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P&{}\approx{}&0.159447590766&{}:{}&0.246863875275&{}:{}&0.593688533959&,\\ A^{\prime\prime}&{}\approx{}&0.113238597558&{}:{}&0.260435106540&{}:{}&0.626326295902&,\\B^{\prime\prime}&{}\approx{}&0.174423075766&{}:{}&0.176128540332&{}:{}&0.649448383901&,\\C^{\prime\prime}&{}\approx{}&0.249574943921&{}:{}&0.386403065307&{}:{}&0.364021990772&. \end{alignedat} \]
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First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-&{}\approx{}&0.141123855203&{}:{}&0.221948657777&{}:{}&0.636927487020&,\\ A^{\prime\prime\prime}&{}\approx{}&0.076951751851&{}:{}&0.238531854658&{}:{}&0.684516393491&,\\B^{\prime\prime\prime}&{}\approx{}&0.159380182962&{}:{}&0.121296926738&{}:{}&0.719322890300&,\\C^{\prime\prime\prime}&{}\approx{}&0.272329471274&{}:{}&0.428298678033&{}:{}&0.299371850692&. \end{alignedat} \]
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Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+&{}\approx{}&0.170431169963&{}:{}&0.261798508208&{}:{}&0.567770321829&,\\ A^\prime&{}\approx{}&0.349607047620&{}:{}&0.266069844155&{}:{}&0.384323108224&,\\B^\prime&{}\approx{}&0.159580615309&{}:{}&0.494661384854&{}:{}&0.345757999837&,\\C^\prime&{}\approx{}&0.087754730048&{}:{}&0.131632095073&{}:{}&0.780613174879&,\\ A^{\prime\prime}&{}\approx{}&0.113238597558&{}:{}&0.260435106540&{}:{}&0.626326295902&,\\B^{\prime\prime}&{}\approx{}&0.174423075766&{}:{}&0.176128540332&{}:{}&0.649448383901&,\\C^{\prime\prime}&{}\approx{}&0.249574943921&{}:{}&0.386403065307&{}:{}&0.364021990772&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q&{}\approx{}&0.124506936676&{}:{}&0.195962661716&{}:{}&0.679530401609&,\\ A^*&{}\approx{}&0.000000000000&{}:{}&0.223831198584&{}:{}&0.776168801416&,\\B^*&{}\approx{}&0.154852182538&{}:{}&0.000000000000&{}:{}&0.845147817462&,\\C^*&{}\approx{}&0.388514034719&{}:{}&0.611485965281&{}:{}&0.000000000000&. \end{alignedat} \]
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Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.186419499351&{}:{}&0.284090530142&{}:{}&0.529489970507&,\\ A^{\prime\prime}&{}\approx{}&0.113238597558&{}:{}&0.260435106540&{}:{}&0.626326295902&,\\B^{\prime\prime}&{}\approx{}&0.174423075766&{}:{}&0.176128540332&{}:{}&0.649448383901&,\\C^{\prime\prime}&{}\approx{}&0.249574943921&{}:{}&0.386403065307&{}:{}&0.364021990772&,\\ A^*&{}\approx{}&0.000000000000&{}:{}&0.223831198584&{}:{}&0.776168801416&,\\B^*&{}\approx{}&0.154852182538&{}:{}&0.000000000000&{}:{}&0.845147817462&,\\C^*&{}\approx{}&0.388514034719&{}:{}&0.611485965281&{}:{}&0.000000000000&. \end{alignedat} \]
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Hiroyasu Kamo