Derousseau's Generalization of the Malfatti circles

Isosceles Triangle with 120° Top Angle

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


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7(222)

Malfatti circles

7 (222)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.267949192431&{}:{}&0.267949192431&{}:{}&0.464101615138&, \\ P_{\mathbf{7}}&{}\approx{}&0.497997059411&{}:{}&0.497997059411&{}:{}&0.004005881179&, \\ P^-_{\mathbf{7}}&{}\approx{}&0.000000000000&{}:{}&0.000000000000&{}:{}&1.000000000000&, \\ P^+_{\mathbf{7}}&{}\approx{}&0.348426301666&{}:{}&0.348426301666&{}:{}&0.303147396668&, \\ Q_{\mathbf{7}}&{}\approx{}&0.448827229421&{}:{}&0.448827229421&{}:{}&0.102345541159&, \\ I^\prime_{\mathbf{7}}&{}\approx{}&0.472135417461&{}:{}&0.472135417461&{}:{}&0.055729165078&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{7}}\)
\(P^-_{\mathbf{7}}\)
\(P^+_{\mathbf{7}}\)
\(Q_{\mathbf{7}}\)
\(I^\prime_{\mathbf{7}}\)
7 (222)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7}}&{}\approx{}&-0.000000000000&{}:{}&0.366025403784&{}:{}&0.633974596216&,\\B^\prime_{\mathbf{7}}&{}\approx{}&0.366025403784&{}:{}&-0.000000000000&{}:{}&0.633974596216&,\\C^\prime_{\mathbf{7}}&{}\approx{}&3.623277963258&{}:{}&3.623277963258&{}:{}&-6.246555926517&, \\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.664883894507&{}:{}&0.332441947254&{}:{}&0.002674158239&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.332441947254&{}:{}&0.664883894507&{}:{}&0.002674158239&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.183012701892&{}:{}&0.183012701892&{}:{}&0.633974596216&, \\ A^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&-1.381202455099&{}:{}&0.000000000000&{}:{}&2.381202455099&,\\B^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&0.000000000000&{}:{}&-1.381202455099&{}:{}&2.381202455099&,\\C^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&-0.000000000000&{}:{}&-0.000000000000&{}:{}&1.000000000000&, \\ A^*_{\mathbf{7}}&{}\approx{}&0.000000000000&{}:{}&0.814313139868&{}:{}&0.185686860132&,\\B^*_{\mathbf{7}}&{}\approx{}&0.814313139868&{}:{}&0.000000000000&{}:{}&0.185686860132&,\\C^*_{\mathbf{7}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7}}}{B^\prime_{\mathbf{7}}}{C^\prime_{\mathbf{7}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7}}}{B^{\prime\prime}_{\mathbf{7}}}{C^{\prime\prime}_{\mathbf{7}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7}}}{B^{\prime\prime\prime}_{\mathbf{7}}}{C^{\prime\prime\prime}_{\mathbf{7}}}\)
\(\triangle{A^*_{\mathbf{7}}}{B^*_{\mathbf{7}}}{C^*_{\mathbf{7}}}\)
7 (222)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7}}}}&{}\approx{}&1.366025403784&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7}}}}&{}\approx{}&1.366025403784&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7}}}}&{}\approx{}&13.522257448825&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.267949192431&{}:{}&0.267949192431&{}:{}&0.464101615138&,\\ A^\prime_{\mathbf{7}}&{}\approx{}&-0.000000000000&{}:{}&0.366025403784&{}:{}&0.633974596216&,\\B^\prime_{\mathbf{7}}&{}\approx{}&0.366025403784&{}:{}&-0.000000000000&{}:{}&0.633974596216&,\\C^\prime_{\mathbf{7}}&{}\approx{}&3.623277963258&{}:{}&3.623277963258&{}:{}&-6.246555926517&. \end{alignedat} \]
7 (222)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7}}&{}\approx{}&0.497997059411&{}:{}&0.497997059411&{}:{}&0.004005881179&,\\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.664883894507&{}:{}&0.332441947254&{}:{}&0.002674158239&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.332441947254&{}:{}&0.664883894507&{}:{}&0.002674158239&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.183012701892&{}:{}&0.183012701892&{}:{}&0.633974596216&. \end{alignedat} \]
7 (222)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7}}&{}\approx{}&0.000000000000&{}:{}&0.000000000000&{}:{}&1.000000000000&,\\ A^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&-1.381202455099&{}:{}&0.000000000000&{}:{}&2.381202455099&,\\B^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&0.000000000000&{}:{}&-1.381202455099&{}:{}&2.381202455099&,\\C^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&-0.000000000000&{}:{}&-0.000000000000&{}:{}&1.000000000000&. \end{alignedat} \]
7 (222)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7}}&{}\approx{}&0.348426301666&{}:{}&0.348426301666&{}:{}&0.303147396668&,\\ A^\prime_{\mathbf{7}}&{}\approx{}&-0.000000000000&{}:{}&0.366025403784&{}:{}&0.633974596216&,\\B^\prime_{\mathbf{7}}&{}\approx{}&0.366025403784&{}:{}&-0.000000000000&{}:{}&0.633974596216&,\\C^\prime_{\mathbf{7}}&{}\approx{}&3.623277963258&{}:{}&3.623277963258&{}:{}&-6.246555926517&,\\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.664883894507&{}:{}&0.332441947254&{}:{}&0.002674158239&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.332441947254&{}:{}&0.664883894507&{}:{}&0.002674158239&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.183012701892&{}:{}&0.183012701892&{}:{}&0.633974596216&, \end{alignedat} \]
7 (222)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7}}&{}\approx{}&0.448827229421&{}:{}&0.448827229421&{}:{}&0.102345541159&,\\ A^*_{\mathbf{7}}&{}\approx{}&0.000000000000&{}:{}&0.814313139868&{}:{}&0.185686860132&,\\B^*_{\mathbf{7}}&{}\approx{}&0.814313139868&{}:{}&0.000000000000&{}:{}&0.185686860132&,\\C^*_{\mathbf{7}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
7 (222)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7}}&{}\approx{}&0.472135417461&{}:{}&0.472135417461&{}:{}&0.055729165078&,\\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.664883894507&{}:{}&0.332441947254&{}:{}&0.002674158239&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.332441947254&{}:{}&0.664883894507&{}:{}&0.002674158239&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.183012701892&{}:{}&0.183012701892&{}:{}&0.633974596216&,\\ A^*_{\mathbf{7}}&{}\approx{}&0.000000000000&{}:{}&0.814313139868&{}:{}&0.185686860132&,\\B^*_{\mathbf{7}}&{}\approx{}&0.814313139868&{}:{}&0.000000000000&{}:{}&0.185686860132&,\\C^*_{\mathbf{7}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
7 (222)