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 > 6 (220)

6(220)

Malfatti circles

6 (220)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.267949192431&{}:{}&0.267949192431&{}:{}&0.464101615138&, \\ P_{\mathbf{6}}&{}\approx{}&0.499776655765&{}:{}&0.499776655765&{}:{}&0.000446688470&, \\ P^-_{\mathbf{6}}&{}\approx{}&0.880812611974&{}:{}&0.880812611974&{}:{}&-0.761625223947&, \\ P^+_{\mathbf{6}}&{}\approx{}&0.410899852618&{}:{}&0.410899852618&{}:{}&0.178200294764&, \\ Q_{\mathbf{6}}&{}\approx{}&0.519753114443&{}:{}&0.519753114443&{}:{}&-0.039506228886&, \\ I^\prime_{\mathbf{6}}&{}\approx{}&0.490353408147&{}:{}&0.490353408147&{}:{}&0.019293183706&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{6}}\)
\(P^-_{\mathbf{6}}\)
\(P^+_{\mathbf{6}}\)
\(Q_{\mathbf{6}}\)
\(I^\prime_{\mathbf{6}}\)
6 (220)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6}}&{}\approx{}&0.422649730810&{}:{}&0.211324865405&{}:{}&0.366025403784&,\\B^\prime_{\mathbf{6}}&{}\approx{}&0.211324865405&{}:{}&0.422649730810&{}:{}&0.366025403784&,\\C^\prime_{\mathbf{6}}&{}\approx{}&3.695074732983&{}:{}&3.695074732983&{}:{}&-6.390149465966&, \\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.399785608688&{}:{}&0.599678413032&{}:{}&0.000535978280&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.599678413032&{}:{}&0.399785608688&{}:{}&0.000535978280&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.316987298108&{}:{}&0.316987298108&{}:{}&0.366025403784&, \\ A^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&0.787010845008&{}:{}&1.574021690017&{}:{}&-1.361032535025&,\\B^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&1.574021690017&{}:{}&0.787010845008&{}:{}&-1.361032535025&,\\C^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&0.348915260374&{}:{}&0.348915260374&{}:{}&0.302169479252&, \\ A^*_{\mathbf{6}}&{}\approx{}&0.000000000000&{}:{}&1.082262332300&{}:{}&-0.082262332300&,\\B^*_{\mathbf{6}}&{}\approx{}&1.082262332300&{}:{}&0.000000000000&{}:{}&-0.082262332300&,\\C^*_{\mathbf{6}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6}}}{B^\prime_{\mathbf{6}}}{C^\prime_{\mathbf{6}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6}}}{B^{\prime\prime}_{\mathbf{6}}}{C^{\prime\prime}_{\mathbf{6}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6}}}{B^{\prime\prime\prime}_{\mathbf{6}}}{C^{\prime\prime\prime}_{\mathbf{6}}}\)
\(\triangle{A^*_{\mathbf{6}}}{B^*_{\mathbf{6}}}{C^*_{\mathbf{6}}}\)
6 (220)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6}}}}&{}\approx{}&0.788675134595&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6}}}}&{}\approx{}&0.788675134595&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6}}}}&{}\approx{}&13.790206641256&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.267949192431&{}:{}&0.267949192431&{}:{}&0.464101615138&,\\ A^\prime_{\mathbf{6}}&{}\approx{}&0.422649730810&{}:{}&0.211324865405&{}:{}&0.366025403784&,\\B^\prime_{\mathbf{6}}&{}\approx{}&0.211324865405&{}:{}&0.422649730810&{}:{}&0.366025403784&,\\C^\prime_{\mathbf{6}}&{}\approx{}&3.695074732983&{}:{}&3.695074732983&{}:{}&-6.390149465966&. \end{alignedat} \]
6 (220)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6}}&{}\approx{}&0.499776655765&{}:{}&0.499776655765&{}:{}&0.000446688470&,\\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.399785608688&{}:{}&0.599678413032&{}:{}&0.000535978280&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.599678413032&{}:{}&0.399785608688&{}:{}&0.000535978280&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.316987298108&{}:{}&0.316987298108&{}:{}&0.366025403784&. \end{alignedat} \]
6 (220)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6}}&{}\approx{}&0.880812611974&{}:{}&0.880812611974&{}:{}&-0.761625223947&,\\ A^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&0.787010845008&{}:{}&1.574021690017&{}:{}&-1.361032535025&,\\B^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&1.574021690017&{}:{}&0.787010845008&{}:{}&-1.361032535025&,\\C^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&0.348915260374&{}:{}&0.348915260374&{}:{}&0.302169479252&. \end{alignedat} \]
6 (220)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6}}&{}\approx{}&0.410899852618&{}:{}&0.410899852618&{}:{}&0.178200294764&,\\ A^\prime_{\mathbf{6}}&{}\approx{}&0.422649730810&{}:{}&0.211324865405&{}:{}&0.366025403784&,\\B^\prime_{\mathbf{6}}&{}\approx{}&0.211324865405&{}:{}&0.422649730810&{}:{}&0.366025403784&,\\C^\prime_{\mathbf{6}}&{}\approx{}&3.695074732983&{}:{}&3.695074732983&{}:{}&-6.390149465966&,\\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.399785608688&{}:{}&0.599678413032&{}:{}&0.000535978280&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.599678413032&{}:{}&0.399785608688&{}:{}&0.000535978280&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.316987298108&{}:{}&0.316987298108&{}:{}&0.366025403784&, \end{alignedat} \]
6 (220)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6}}&{}\approx{}&0.519753114443&{}:{}&0.519753114443&{}:{}&-0.039506228886&,\\ A^*_{\mathbf{6}}&{}\approx{}&0.000000000000&{}:{}&1.082262332300&{}:{}&-0.082262332300&,\\B^*_{\mathbf{6}}&{}\approx{}&1.082262332300&{}:{}&0.000000000000&{}:{}&-0.082262332300&,\\C^*_{\mathbf{6}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
6 (220)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6}}&{}\approx{}&0.490353408147&{}:{}&0.490353408147&{}:{}&0.019293183706&,\\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.399785608688&{}:{}&0.599678413032&{}:{}&0.000535978280&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.599678413032&{}:{}&0.399785608688&{}:{}&0.000535978280&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.316987298108&{}:{}&0.316987298108&{}:{}&0.366025403784&,\\ A^*_{\mathbf{6}}&{}\approx{}&0.000000000000&{}:{}&1.082262332300&{}:{}&-0.082262332300&,\\B^*_{\mathbf{6}}&{}\approx{}&1.082262332300&{}:{}&0.000000000000&{}:{}&-0.082262332300&,\\C^*_{\mathbf{6}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
6 (220)