Derousseau's Generalization of the Malfatti circles

Test Case in ETC

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

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

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Malfatti circles

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Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.214285714286&{}:{}&0.321428571429&{}:{}&0.464285714286&, \\ P_{\mathbf{1}}&{}\approx{}&0.026489928890&{}:{}&0.041012890004&{}:{}&0.932497181107&, \\ P^-_{\mathbf{1}}&{}\approx{}&-0.001061519089&{}:{}&-0.000126787561&{}:{}&1.001188306649&, \\ P^+_{\mathbf{1}}&{}\approx{}&0.047791180025&{}:{}&0.072819806435&{}:{}&0.879389013540&, \\ Q_{\mathbf{1}}&{}\approx{}&-0.070385374478&{}:{}&-0.110780215921&{}:{}&1.181165590399&, \\ I^\prime_{\mathbf{1}}&{}\approx{}&0.088831623580&{}:{}&0.135373301206&{}:{}&0.775795075214&, \end{alignedat} \]
\(I\) Incenter
\(P_{\mathbf{1}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{1}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{1}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{1}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{1}}\) Radical center of the Malfatti circles
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Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1}}&{}\approx{}&-0.009294423249&{}:{}&0.412893173147&{}:{}&0.596401250102&,\\B^\prime_{\mathbf{1}}&{}\approx{}&0.316192166234&{}:{}&-0.001275193075&{}:{}&0.685083026841&,\\C^\prime_{\mathbf{1}}&{}\approx{}&0.029114908048&{}:{}&0.043672362073&{}:{}&0.927212729879&, \\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.053320111467&{}:{}&0.039882564433&{}:{}&0.906797324100&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.025440898308&{}:{}&0.078989843797&{}:{}&0.895569257895&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.142186646282&{}:{}&0.220139710766&{}:{}&0.637673642951&, \\ A^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&0.030448229362&{}:{}&-0.000122796753&{}:{}&0.969674567391&,\\B^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&-0.001013592918&{}:{}&0.045027602578&{}:{}&0.955985990340&,\\C^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&-0.015860879755&{}:{}&-0.001894419302&{}:{}&1.017755299057&, \\ A^*_{\mathbf{1}}&{}\approx{}&0.000000000000&{}:{}&-0.103495636770&{}:{}&1.103495636770&,\\B^*_{\mathbf{1}}&{}\approx{}&-0.063365707697&{}:{}&0.000000000000&{}:{}&1.063365707697&,\\C^*_{\mathbf{1}}&{}\approx{}&0.388514034719&{}:{}&0.611485965281&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1}}}{B^\prime_{\mathbf{1}}}{C^\prime_{\mathbf{1}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1}}}{B^{\prime\prime}_{\mathbf{1}}}{C^{\prime\prime}_{\mathbf{1}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1}}}{B^{\prime\prime\prime}_{\mathbf{1}}}{C^{\prime\prime\prime}_{\mathbf{1}}}\)
\(\triangle{A^*_{\mathbf{1}}}{B^*_{\mathbf{1}}}{C^*_{\mathbf{1}}}\)
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Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1}}}}&{}\approx{}&1.284556538681&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1}}}}&{}\approx{}&1.475563442427&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1}}}}&{}\approx{}&0.135869570892&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.214285714286&{}:{}&0.321428571429&{}:{}&0.464285714286&,\\ A^\prime_{\mathbf{1}}&{}\approx{}&-0.009294423249&{}:{}&0.412893173147&{}:{}&0.596401250102&,\\B^\prime_{\mathbf{1}}&{}\approx{}&0.316192166234&{}:{}&-0.001275193075&{}:{}&0.685083026841&,\\C^\prime_{\mathbf{1}}&{}\approx{}&0.029114908048&{}:{}&0.043672362073&{}:{}&0.927212729879&. \end{alignedat} \]
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First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1}}&{}\approx{}&0.026489928890&{}:{}&0.041012890004&{}:{}&0.932497181107&,\\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.053320111467&{}:{}&0.039882564433&{}:{}&0.906797324100&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.025440898308&{}:{}&0.078989843797&{}:{}&0.895569257895&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.142186646282&{}:{}&0.220139710766&{}:{}&0.637673642951&. \end{alignedat} \]
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First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1}}&{}\approx{}&-0.001061519089&{}:{}&-0.000126787561&{}:{}&1.001188306649&,\\ A^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&0.030448229362&{}:{}&-0.000122796753&{}:{}&0.969674567391&,\\B^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&-0.001013592918&{}:{}&0.045027602578&{}:{}&0.955985990340&,\\C^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&-0.015860879755&{}:{}&-0.001894419302&{}:{}&1.017755299057&. \end{alignedat} \]
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Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1}}&{}\approx{}&0.047791180025&{}:{}&0.072819806435&{}:{}&0.879389013540&,\\ A^\prime_{\mathbf{1}}&{}\approx{}&-0.009294423249&{}:{}&0.412893173147&{}:{}&0.596401250102&,\\B^\prime_{\mathbf{1}}&{}\approx{}&0.316192166234&{}:{}&-0.001275193075&{}:{}&0.685083026841&,\\C^\prime_{\mathbf{1}}&{}\approx{}&0.029114908048&{}:{}&0.043672362073&{}:{}&0.927212729879&,\\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.053320111467&{}:{}&0.039882564433&{}:{}&0.906797324100&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.025440898308&{}:{}&0.078989843797&{}:{}&0.895569257895&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.142186646282&{}:{}&0.220139710766&{}:{}&0.637673642951&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1}}&{}\approx{}&-0.070385374478&{}:{}&-0.110780215921&{}:{}&1.181165590399&,\\ A^*_{\mathbf{1}}&{}\approx{}&0.000000000000&{}:{}&-0.103495636770&{}:{}&1.103495636770&,\\B^*_{\mathbf{1}}&{}\approx{}&-0.063365707697&{}:{}&0.000000000000&{}:{}&1.063365707697&,\\C^*_{\mathbf{1}}&{}\approx{}&0.388514034719&{}:{}&0.611485965281&{}:{}&0.000000000000&. \end{alignedat} \]
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Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1}}&{}\approx{}&0.088831623580&{}:{}&0.135373301206&{}:{}&0.775795075214&,\\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.053320111467&{}:{}&0.039882564433&{}:{}&0.906797324100&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.025440898308&{}:{}&0.078989843797&{}:{}&0.895569257895&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.142186646282&{}:{}&0.220139710766&{}:{}&0.637673642951&,\\ A^*_{\mathbf{1}}&{}\approx{}&0.000000000000&{}:{}&-0.103495636770&{}:{}&1.103495636770&,\\B^*_{\mathbf{1}}&{}\approx{}&-0.063365707697&{}:{}&0.000000000000&{}:{}&1.063365707697&,\\C^*_{\mathbf{1}}&{}\approx{}&0.388514034719&{}:{}&0.611485965281&{}:{}&0.000000000000&. \end{alignedat} \]
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