Derousseau's Generalization of the Malfatti circles

Isosceles Triangle with 120° Top Angle

\(A=30\degree\), \(B=30\degree\), \(C=120\degree\).

[Top] > Isosceles Triangle with 120° Top Angle > 0 (000)

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Malfatti circles

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Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.267949192431&{}:{}&0.267949192431&{}:{}&0.464101615138&, \\ P&{}\approx{}&0.200997996569&{}:{}&0.200997996569&{}:{}&0.598004006862&, \\ P^-&{}\approx{}&0.178675585363&{}:{}&0.178675585363&{}:{}&0.642648829273&, \\ P^+&{}\approx{}&0.214390160088&{}:{}&0.214390160088&{}:{}&0.571219679824&, \\ Q&{}\approx{}&0.156606998336&{}:{}&0.156606998336&{}:{}&0.686786003328&, \\ I^\prime&{}\approx{}&0.234188987528&{}:{}&0.234188987528&{}:{}&0.531622024945&, \end{alignedat} \]
\(I\) Incenter
\(P\) First Ajima-Malfatti point
\(P^-\) First Malfatti-Rabinowitz point
\(P^+\) Gergonne point of the Malfatti triangle
\(Q\) Second Malfatti-Rabinowitz point
\(I^\prime\) Radical center of the Malfatti circles
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Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.422649730810&{}:{}&0.211324865405&{}:{}&0.366025403784&,\\B^\prime&{}\approx{}&0.211324865405&{}:{}&0.422649730810&{}:{}&0.366025403784&,\\C^\prime&{}\approx{}&0.108772844311&{}:{}&0.108772844311&{}:{}&0.782454311379&, \\ A^{\prime\prime}&{}\approx{}&0.143621188068&{}:{}&0.215431782103&{}:{}&0.640947029829&,\\B^{\prime\prime}&{}\approx{}&0.215431782103&{}:{}&0.143621188068&{}:{}&0.640947029829&,\\C^{\prime\prime}&{}\approx{}&0.316987298108&{}:{}&0.316987298108&{}:{}&0.366025403784&, \\ A^{\prime\prime\prime}&{}\approx{}&0.098102009685&{}:{}&0.196204019369&{}:{}&0.705693970946&,\\B^{\prime\prime\prime}&{}\approx{}&0.196204019369&{}:{}&0.098102009685&{}:{}&0.705693970946&,\\C^{\prime\prime\prime}&{}\approx{}&0.348915260374&{}:{}&0.348915260374&{}:{}&0.302169479252&, \\ A^*&{}\approx{}&0.000000000000&{}:{}&0.185686860132&{}:{}&0.814313139868&,\\B^*&{}\approx{}&0.185686860132&{}:{}&0.000000000000&{}:{}&0.814313139868&,\\C^*&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime}{B^\prime}{C^\prime}\)
\(\triangle{A^{\prime\prime}}{B^{\prime\prime}}{C^{\prime\prime}}\)
\(\triangle{A^{\prime\prime\prime}}{B^{\prime\prime\prime}}{C^{\prime\prime\prime}}\)
\(\triangle{A^*}{B^*}{C^*}\)
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Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime}}&{}\approx{}&0.788675134595&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime}}&{}\approx{}&0.788675134595&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime}}&{}\approx{}&0.405945781451&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.267949192431&{}:{}&0.267949192431&{}:{}&0.464101615138&,\\ A^\prime&{}\approx{}&0.422649730810&{}:{}&0.211324865405&{}:{}&0.366025403784&,\\B^\prime&{}\approx{}&0.211324865405&{}:{}&0.422649730810&{}:{}&0.366025403784&,\\C^\prime&{}\approx{}&0.108772844311&{}:{}&0.108772844311&{}:{}&0.782454311379&. \end{alignedat} \]
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First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P&{}\approx{}&0.200997996569&{}:{}&0.200997996569&{}:{}&0.598004006862&,\\ A^{\prime\prime}&{}\approx{}&0.143621188068&{}:{}&0.215431782103&{}:{}&0.640947029829&,\\B^{\prime\prime}&{}\approx{}&0.215431782103&{}:{}&0.143621188068&{}:{}&0.640947029829&,\\C^{\prime\prime}&{}\approx{}&0.316987298108&{}:{}&0.316987298108&{}:{}&0.366025403784&. \end{alignedat} \]
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First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-&{}\approx{}&0.178675585363&{}:{}&0.178675585363&{}:{}&0.642648829273&,\\ A^{\prime\prime\prime}&{}\approx{}&0.098102009685&{}:{}&0.196204019369&{}:{}&0.705693970946&,\\B^{\prime\prime\prime}&{}\approx{}&0.196204019369&{}:{}&0.098102009685&{}:{}&0.705693970946&,\\C^{\prime\prime\prime}&{}\approx{}&0.348915260374&{}:{}&0.348915260374&{}:{}&0.302169479252&. \end{alignedat} \]
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Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+&{}\approx{}&0.214390160088&{}:{}&0.214390160088&{}:{}&0.571219679824&,\\ A^\prime&{}\approx{}&0.422649730810&{}:{}&0.211324865405&{}:{}&0.366025403784&,\\B^\prime&{}\approx{}&0.211324865405&{}:{}&0.422649730810&{}:{}&0.366025403784&,\\C^\prime&{}\approx{}&0.108772844311&{}:{}&0.108772844311&{}:{}&0.782454311379&,\\ A^{\prime\prime}&{}\approx{}&0.143621188068&{}:{}&0.215431782103&{}:{}&0.640947029829&,\\B^{\prime\prime}&{}\approx{}&0.215431782103&{}:{}&0.143621188068&{}:{}&0.640947029829&,\\C^{\prime\prime}&{}\approx{}&0.316987298108&{}:{}&0.316987298108&{}:{}&0.366025403784&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q&{}\approx{}&0.156606998336&{}:{}&0.156606998336&{}:{}&0.686786003328&,\\ A^*&{}\approx{}&0.000000000000&{}:{}&0.185686860132&{}:{}&0.814313139868&,\\B^*&{}\approx{}&0.185686860132&{}:{}&0.000000000000&{}:{}&0.814313139868&,\\C^*&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
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Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.234188987528&{}:{}&0.234188987528&{}:{}&0.531622024945&,\\ A^{\prime\prime}&{}\approx{}&0.143621188068&{}:{}&0.215431782103&{}:{}&0.640947029829&,\\B^{\prime\prime}&{}\approx{}&0.215431782103&{}:{}&0.143621188068&{}:{}&0.640947029829&,\\C^{\prime\prime}&{}\approx{}&0.316987298108&{}:{}&0.316987298108&{}:{}&0.366025403784&,\\ A^*&{}\approx{}&0.000000000000&{}:{}&0.185686860132&{}:{}&0.814313139868&,\\B^*&{}\approx{}&0.185686860132&{}:{}&0.000000000000&{}:{}&0.814313139868&,\\C^*&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
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