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 > 7c (332)

7c(332)

Malfatti circles

7c (332)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.732050807569&{}:{}&3.732050807569&{}:{}&-6.464101615138&, \\ P_{\mathbf{7c}}&{}\approx{}&-0.111375854809&{}:{}&-0.111375854809&{}:{}&1.222751709617&, \\ P^-_{\mathbf{7c}}&{}\approx{}&-0.000000000000&{}:{}&-0.000000000000&{}:{}&1.000000000000&, \\ P^+_{\mathbf{7c}}&{}\approx{}&-0.229603791727&{}:{}&-0.229603791727&{}:{}&1.459207583454&, \\ Q_{\mathbf{7c}}&{}\approx{}&0.234893275893&{}:{}&0.234893275893&{}:{}&0.530213448213&, \\ I^\prime_{\mathbf{7c}}&{}\approx{}&-0.423607280037&{}:{}&-0.423607280037&{}:{}&1.847214560075&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{7c}}\)
\(P^-_{\mathbf{7c}}\)
\(P^+_{\mathbf{7c}}\)
\(Q_{\mathbf{7c}}\)
\(I^\prime_{\mathbf{7c}}\)
7c (332)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&-1.366025403784&{}:{}&2.366025403784&,\\B^\prime_{\mathbf{7c}}&{}\approx{}&-1.366025403784&{}:{}&0.000000000000&{}:{}&2.366025403784&,\\C^\prime_{\mathbf{7c}}&{}\approx{}&-0.137996589020&{}:{}&-0.137996589020&{}:{}&1.275993178039&, \\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.250670332134&{}:{}&-0.125335166067&{}:{}&1.376005498201&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.125335166067&{}:{}&-0.250670332134&{}:{}&1.376005498201&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.683012701892&{}:{}&-0.683012701892&{}:{}&2.366025403784&, \\ A^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.121262743976&{}:{}&-0.000000000000&{}:{}&1.121262743976&,\\B^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&-0.121262743976&{}:{}&1.121262743976&,\\C^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&0.000000000000&{}:{}&1.000000000000&, \\ A^*_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&0.307007203691&{}:{}&0.692992796309&,\\B^*_{\mathbf{7c}}&{}\approx{}&0.307007203691&{}:{}&0.000000000000&{}:{}&0.692992796309&,\\C^*_{\mathbf{7c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7c}}}{B^\prime_{\mathbf{7c}}}{C^\prime_{\mathbf{7c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7c}}}{B^{\prime\prime}_{\mathbf{7c}}}{C^{\prime\prime}_{\mathbf{7c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7c}}}{B^{\prime\prime\prime}_{\mathbf{7c}}}{C^{\prime\prime\prime}_{\mathbf{7c}}}\)
\(\triangle{A^*_{\mathbf{7c}}}{B^*_{\mathbf{7c}}}{C^*_{\mathbf{7c}}}\)
7c (332)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7c}}}}&{}\approx{}&-0.366025403784&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7c}}}}&{}\approx{}&-0.366025403784&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7c}}}}&{}\approx{}&-0.036976074586&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.732050807569&{}:{}&3.732050807569&{}:{}&-6.464101615138&,\\ A^\prime_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&-1.366025403784&{}:{}&2.366025403784&,\\B^\prime_{\mathbf{7c}}&{}\approx{}&-1.366025403784&{}:{}&0.000000000000&{}:{}&2.366025403784&,\\C^\prime_{\mathbf{7c}}&{}\approx{}&-0.137996589020&{}:{}&-0.137996589020&{}:{}&1.275993178039&. \end{alignedat} \]
7c (332)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7c}}&{}\approx{}&-0.111375854809&{}:{}&-0.111375854809&{}:{}&1.222751709617&,\\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.250670332134&{}:{}&-0.125335166067&{}:{}&1.376005498201&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.125335166067&{}:{}&-0.250670332134&{}:{}&1.376005498201&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.683012701892&{}:{}&-0.683012701892&{}:{}&2.366025403784&. \end{alignedat} \]
7c (332)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7c}}&{}\approx{}&-0.000000000000&{}:{}&-0.000000000000&{}:{}&1.000000000000&,\\ A^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.121262743976&{}:{}&-0.000000000000&{}:{}&1.121262743976&,\\B^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&-0.121262743976&{}:{}&1.121262743976&,\\C^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&0.000000000000&{}:{}&1.000000000000&. \end{alignedat} \]
7c (332)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7c}}&{}\approx{}&-0.229603791727&{}:{}&-0.229603791727&{}:{}&1.459207583454&,\\ A^\prime_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&-1.366025403784&{}:{}&2.366025403784&,\\B^\prime_{\mathbf{7c}}&{}\approx{}&-1.366025403784&{}:{}&0.000000000000&{}:{}&2.366025403784&,\\C^\prime_{\mathbf{7c}}&{}\approx{}&-0.137996589020&{}:{}&-0.137996589020&{}:{}&1.275993178039&,\\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.250670332134&{}:{}&-0.125335166067&{}:{}&1.376005498201&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.125335166067&{}:{}&-0.250670332134&{}:{}&1.376005498201&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.683012701892&{}:{}&-0.683012701892&{}:{}&2.366025403784&, \end{alignedat} \]
7c (332)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7c}}&{}\approx{}&0.234893275893&{}:{}&0.234893275893&{}:{}&0.530213448213&,\\ A^*_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&0.307007203691&{}:{}&0.692992796309&,\\B^*_{\mathbf{7c}}&{}\approx{}&0.307007203691&{}:{}&0.000000000000&{}:{}&0.692992796309&,\\C^*_{\mathbf{7c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
7c (332)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7c}}&{}\approx{}&-0.423607280037&{}:{}&-0.423607280037&{}:{}&1.847214560075&,\\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.250670332134&{}:{}&-0.125335166067&{}:{}&1.376005498201&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.125335166067&{}:{}&-0.250670332134&{}:{}&1.376005498201&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.683012701892&{}:{}&-0.683012701892&{}:{}&2.366025403784&,\\ A^*_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&0.307007203691&{}:{}&0.692992796309&,\\B^*_{\mathbf{7c}}&{}\approx{}&0.307007203691&{}:{}&0.000000000000&{}:{}&0.692992796309&,\\C^*_{\mathbf{7c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
7c (332)