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 > 6c (330)

6c(330)

Malfatti circles

6c (330)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.732050807569&{}:{}&3.732050807569&{}:{}&-6.464101615138&, \\ P_{\mathbf{6c}}&{}\approx{}&0.634070866196&{}:{}&0.634070866196&{}:{}&-0.268141732392&, \\ P^-_{\mathbf{6c}}&{}\approx{}&0.448090591912&{}:{}&0.448090591912&{}:{}&0.103818816175&, \\ P^+_{\mathbf{6c}}&{}\approx{}&0.800114970091&{}:{}&0.800114970091&{}:{}&-0.600229940182&, \\ Q_{\mathbf{6c}}&{}\approx{}&0.409330032484&{}:{}&0.409330032484&{}:{}&0.181339935031&, \\ I^\prime_{\mathbf{6c}}&{}\approx{}&0.874005200027&{}:{}&0.874005200027&{}:{}&-0.748010400053&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{6c}}\)
\(P^-_{\mathbf{6c}}\)
\(P^+_{\mathbf{6c}}\)
\(Q_{\mathbf{6c}}\)
\(I^\prime_{\mathbf{6c}}\)
6c (330)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6c}}&{}\approx{}&1.577350269190&{}:{}&0.788675134595&{}:{}&-1.366025403784&,\\B^\prime_{\mathbf{6c}}&{}\approx{}&0.788675134595&{}:{}&1.577350269190&{}:{}&-1.366025403784&,\\C^\prime_{\mathbf{6c}}&{}\approx{}&0.405945781451&{}:{}&0.405945781451&{}:{}&0.188108437098&, \\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.536001570915&{}:{}&0.804002356372&{}:{}&-0.340003927286&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.804002356372&{}:{}&0.536001570915&{}:{}&-0.340003927286&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&1.183012701892&{}:{}&1.183012701892&{}:{}&-1.366025403784&, \\ A^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.288735018666&{}:{}&0.577470037332&{}:{}&0.133794944001&,\\B^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.577470037332&{}:{}&0.288735018666&{}:{}&0.133794944001&,\\C^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.881853970395&{}:{}&0.881853970395&{}:{}&-0.763707940790&, \\ A^*_{\mathbf{6c}}&{}\approx{}&0.000000000000&{}:{}&0.692992796309&{}:{}&0.307007203691&,\\B^*_{\mathbf{6c}}&{}\approx{}&0.692992796309&{}:{}&0.000000000000&{}:{}&0.307007203691&,\\C^*_{\mathbf{6c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6c}}}{B^\prime_{\mathbf{6c}}}{C^\prime_{\mathbf{6c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6c}}}{B^{\prime\prime}_{\mathbf{6c}}}{C^{\prime\prime}_{\mathbf{6c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6c}}}{B^{\prime\prime\prime}_{\mathbf{6c}}}{C^{\prime\prime\prime}_{\mathbf{6c}}}\)
\(\triangle{A^*_{\mathbf{6c}}}{B^*_{\mathbf{6c}}}{C^*_{\mathbf{6c}}}\)
6c (330)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6c}}}}&{}\approx{}&0.211324865405&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6c}}}}&{}\approx{}&0.211324865405&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6c}}}}&{}\approx{}&0.108772844311&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.732050807569&{}:{}&3.732050807569&{}:{}&-6.464101615138&,\\ A^\prime_{\mathbf{6c}}&{}\approx{}&1.577350269190&{}:{}&0.788675134595&{}:{}&-1.366025403784&,\\B^\prime_{\mathbf{6c}}&{}\approx{}&0.788675134595&{}:{}&1.577350269190&{}:{}&-1.366025403784&,\\C^\prime_{\mathbf{6c}}&{}\approx{}&0.405945781451&{}:{}&0.405945781451&{}:{}&0.188108437098&. \end{alignedat} \]
6c (330)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6c}}&{}\approx{}&0.634070866196&{}:{}&0.634070866196&{}:{}&-0.268141732392&,\\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.536001570915&{}:{}&0.804002356372&{}:{}&-0.340003927286&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.804002356372&{}:{}&0.536001570915&{}:{}&-0.340003927286&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&1.183012701892&{}:{}&1.183012701892&{}:{}&-1.366025403784&. \end{alignedat} \]
6c (330)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6c}}&{}\approx{}&0.448090591912&{}:{}&0.448090591912&{}:{}&0.103818816175&,\\ A^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.288735018666&{}:{}&0.577470037332&{}:{}&0.133794944001&,\\B^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.577470037332&{}:{}&0.288735018666&{}:{}&0.133794944001&,\\C^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.881853970395&{}:{}&0.881853970395&{}:{}&-0.763707940790&. \end{alignedat} \]
6c (330)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6c}}&{}\approx{}&0.800114970091&{}:{}&0.800114970091&{}:{}&-0.600229940182&,\\ A^\prime_{\mathbf{6c}}&{}\approx{}&1.577350269190&{}:{}&0.788675134595&{}:{}&-1.366025403784&,\\B^\prime_{\mathbf{6c}}&{}\approx{}&0.788675134595&{}:{}&1.577350269190&{}:{}&-1.366025403784&,\\C^\prime_{\mathbf{6c}}&{}\approx{}&0.405945781451&{}:{}&0.405945781451&{}:{}&0.188108437098&,\\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.536001570915&{}:{}&0.804002356372&{}:{}&-0.340003927286&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.804002356372&{}:{}&0.536001570915&{}:{}&-0.340003927286&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&1.183012701892&{}:{}&1.183012701892&{}:{}&-1.366025403784&, \end{alignedat} \]
6c (330)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6c}}&{}\approx{}&0.409330032484&{}:{}&0.409330032484&{}:{}&0.181339935031&,\\ A^*_{\mathbf{6c}}&{}\approx{}&0.000000000000&{}:{}&0.692992796309&{}:{}&0.307007203691&,\\B^*_{\mathbf{6c}}&{}\approx{}&0.692992796309&{}:{}&0.000000000000&{}:{}&0.307007203691&,\\C^*_{\mathbf{6c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
6c (330)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6c}}&{}\approx{}&0.874005200027&{}:{}&0.874005200027&{}:{}&-0.748010400053&,\\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.536001570915&{}:{}&0.804002356372&{}:{}&-0.340003927286&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.804002356372&{}:{}&0.536001570915&{}:{}&-0.340003927286&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&1.183012701892&{}:{}&1.183012701892&{}:{}&-1.366025403784&,\\ A^*_{\mathbf{6c}}&{}\approx{}&0.000000000000&{}:{}&0.692992796309&{}:{}&0.307007203691&,\\B^*_{\mathbf{6c}}&{}\approx{}&0.692992796309&{}:{}&0.000000000000&{}:{}&0.307007203691&,\\C^*_{\mathbf{6c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
6c (330)