Derousseau's Generalization of the Malfatti circles

Isosceles Triangle with 90° Top Angle

\(A=45\degree\), \(B=45\degree\), \(C=90\degree\).


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Malfatti circles

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Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.292893218813&{}:{}&0.292893218813&{}:{}&0.414213562373&, \\ P_{\mathbf{1}}&{}\approx{}&0.015868689191&{}:{}&0.015868689191&{}:{}&0.968262621619&, \\ P^-_{\mathbf{1}}&{}\approx{}&-0.012364438447&{}:{}&-0.012364438447&{}:{}&1.024728876893&, \\ P^+_{\mathbf{1}}&{}\approx{}&0.039321417894&{}:{}&0.039321417894&{}:{}&0.921357164211&, \\ Q_{\mathbf{1}}&{}\approx{}&-0.098647822610&{}:{}&-0.098647822610&{}:{}&1.197295645220&, \\ I^\prime_{\mathbf{1}}&{}\approx{}&0.088579356245&{}:{}&0.088579356245&{}:{}&0.822841287509&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{1}}\)
\(P^-_{\mathbf{1}}\)
\(P^+_{\mathbf{1}}\)
\(Q_{\mathbf{1}}\)
\(I^\prime_{\mathbf{1}}\)
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Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1}}&{}\approx{}&-0.207106781187&{}:{}&0.500000000000&{}:{}&0.707106781187&,\\B^\prime_{\mathbf{1}}&{}\approx{}&0.500000000000&{}:{}&-0.207106781187&{}:{}&0.707106781187&,\\C^\prime_{\mathbf{1}}&{}\approx{}&0.027526354083&{}:{}&0.027526354083&{}:{}&0.944947291833&, \\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.052180054836&{}:{}&0.015283184219&{}:{}&0.932536760946&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.015283184219&{}:{}&0.052180054836&{}:{}&0.932536760946&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.146446609407&{}:{}&0.146446609407&{}:{}&0.707106781187&, \\ A^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&0.028641306985&{}:{}&-0.011863617797&{}:{}&0.983222310812&,\\B^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&-0.011863617797&{}:{}&0.028641306985&{}:{}&0.983222310812&,\\C^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&-0.707106781187&{}:{}&-0.707106781187&{}:{}&2.414213562373&, \\ A^*_{\mathbf{1}}&{}\approx{}&0.000000000000&{}:{}&-0.089790213552&{}:{}&1.089790213552&,\\B^*_{\mathbf{1}}&{}\approx{}&-0.089790213552&{}:{}&0.000000000000&{}:{}&1.089790213552&,\\C^*_{\mathbf{1}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1}}}{B^\prime_{\mathbf{1}}}{C^\prime_{\mathbf{1}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1}}}{B^{\prime\prime}_{\mathbf{1}}}{C^{\prime\prime}_{\mathbf{1}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1}}}{B^{\prime\prime\prime}_{\mathbf{1}}}{C^{\prime\prime\prime}_{\mathbf{1}}}\)
\(\triangle{A^*_{\mathbf{1}}}{B^*_{\mathbf{1}}}{C^*_{\mathbf{1}}}\)
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Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1}}}}&{}\approx{}&1.707106781187&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1}}}}&{}\approx{}&1.707106781187&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1}}}}&{}\approx{}&0.093980851434&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.292893218813&{}:{}&0.292893218813&{}:{}&0.414213562373&,\\ A^\prime_{\mathbf{1}}&{}\approx{}&-0.207106781187&{}:{}&0.500000000000&{}:{}&0.707106781187&,\\B^\prime_{\mathbf{1}}&{}\approx{}&0.500000000000&{}:{}&-0.207106781187&{}:{}&0.707106781187&,\\C^\prime_{\mathbf{1}}&{}\approx{}&0.027526354083&{}:{}&0.027526354083&{}:{}&0.944947291833&. \end{alignedat} \]
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First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1}}&{}\approx{}&0.015868689191&{}:{}&0.015868689191&{}:{}&0.968262621619&,\\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.052180054836&{}:{}&0.015283184219&{}:{}&0.932536760946&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.015283184219&{}:{}&0.052180054836&{}:{}&0.932536760946&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.146446609407&{}:{}&0.146446609407&{}:{}&0.707106781187&. \end{alignedat} \]
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First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1}}&{}\approx{}&-0.012364438447&{}:{}&-0.012364438447&{}:{}&1.024728876893&,\\ A^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&0.028641306985&{}:{}&-0.011863617797&{}:{}&0.983222310812&,\\B^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&-0.011863617797&{}:{}&0.028641306985&{}:{}&0.983222310812&,\\C^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&-0.707106781187&{}:{}&-0.707106781187&{}:{}&2.414213562373&. \end{alignedat} \]
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Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1}}&{}\approx{}&0.039321417894&{}:{}&0.039321417894&{}:{}&0.921357164211&,\\ A^\prime_{\mathbf{1}}&{}\approx{}&-0.207106781187&{}:{}&0.500000000000&{}:{}&0.707106781187&,\\B^\prime_{\mathbf{1}}&{}\approx{}&0.500000000000&{}:{}&-0.207106781187&{}:{}&0.707106781187&,\\C^\prime_{\mathbf{1}}&{}\approx{}&0.027526354083&{}:{}&0.027526354083&{}:{}&0.944947291833&,\\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.052180054836&{}:{}&0.015283184219&{}:{}&0.932536760946&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.015283184219&{}:{}&0.052180054836&{}:{}&0.932536760946&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.146446609407&{}:{}&0.146446609407&{}:{}&0.707106781187&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1}}&{}\approx{}&-0.098647822610&{}:{}&-0.098647822610&{}:{}&1.197295645220&,\\ A^*_{\mathbf{1}}&{}\approx{}&0.000000000000&{}:{}&-0.089790213552&{}:{}&1.089790213552&,\\B^*_{\mathbf{1}}&{}\approx{}&-0.089790213552&{}:{}&0.000000000000&{}:{}&1.089790213552&,\\C^*_{\mathbf{1}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
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Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1}}&{}\approx{}&0.088579356245&{}:{}&0.088579356245&{}:{}&0.822841287509&,\\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.052180054836&{}:{}&0.015283184219&{}:{}&0.932536760946&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.015283184219&{}:{}&0.052180054836&{}:{}&0.932536760946&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.146446609407&{}:{}&0.146446609407&{}:{}&0.707106781187&,\\ A^*_{\mathbf{1}}&{}\approx{}&0.000000000000&{}:{}&-0.089790213552&{}:{}&1.089790213552&,\\B^*_{\mathbf{1}}&{}\approx{}&-0.089790213552&{}:{}&0.000000000000&{}:{}&1.089790213552&,\\C^*_{\mathbf{1}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
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