Derousseau's Generalization of the Malfatti circles

Isosceles Triangle with 108° Top Angle

\(A=36\degree\), \(B=36\degree\), \(C=108\degree\).

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6(220)

Malfatti circles

6 (220)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.276393202250&{}:{}&0.276393202250&{}:{}&0.447213595500&, \\ P_{\mathbf{6}}&{}\approx{}&0.499615939044&{}:{}&0.499615939044&{}:{}&0.000768121912&, \\ P^-_{\mathbf{6}}&{}\approx{}&0.794586950088&{}:{}&0.794586950088&{}:{}&-0.589173900176&, \\ P^+_{\mathbf{6}}&{}\approx{}&0.418643128010&{}:{}&0.418643128010&{}:{}&0.162713743979&, \\ Q_{\mathbf{6}}&{}\approx{}&0.521023534915&{}:{}&0.521023534915&{}:{}&-0.042047069830&, \\ I^\prime_{\mathbf{6}}&{}\approx{}&0.487834409624&{}:{}&0.487834409624&{}:{}&0.024331180752&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{6}}\)
\(P^-_{\mathbf{6}}\)
\(P^+_{\mathbf{6}}\)
\(Q_{\mathbf{6}}\)
\(I^\prime_{\mathbf{6}}\)
6 (220)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6}}&{}\approx{}&0.453848561685&{}:{}&0.208611286432&{}:{}&0.337540151884&,\\B^\prime_{\mathbf{6}}&{}\approx{}&0.208611286432&{}:{}&0.453848561685&{}:{}&0.337540151884&,\\C^\prime_{\mathbf{6}}&{}\approx{}&2.584993206903&{}:{}&2.584993206903&{}:{}&-4.169986413806&, \\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.386066645866&{}:{}&0.612990926709&{}:{}&0.000942427425&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.612990926709&{}:{}&0.386066645866&{}:{}&0.000942427425&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.331229924058&{}:{}&0.331229924058&{}:{}&0.337540151884&, \\ A^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&0.640033295438&{}:{}&1.392437559508&{}:{}&-1.032470854946&,\\B^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&1.392437559508&{}:{}&0.640033295438&{}:{}&-1.032470854946&,\\C^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&0.364467536191&{}:{}&0.364467536191&{}:{}&0.271064927618&, \\ A^*_{\mathbf{6}}&{}\approx{}&0.000000000000&{}:{}&1.087785252292&{}:{}&-0.087785252292&,\\B^*_{\mathbf{6}}&{}\approx{}&1.087785252292&{}:{}&0.000000000000&{}:{}&-0.087785252292&,\\C^*_{\mathbf{6}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6}}}{B^\prime_{\mathbf{6}}}{C^\prime_{\mathbf{6}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6}}}{B^{\prime\prime}_{\mathbf{6}}}{C^{\prime\prime}_{\mathbf{6}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6}}}{B^{\prime\prime\prime}_{\mathbf{6}}}{C^{\prime\prime\prime}_{\mathbf{6}}}\)
\(\triangle{A^*_{\mathbf{6}}}{B^*_{\mathbf{6}}}{C^*_{\mathbf{6}}}\)
6 (220)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6}}}}&{}\approx{}&0.754762724747&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6}}}}&{}\approx{}&0.754762724747&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6}}}}&{}\approx{}&9.352593283263&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.276393202250&{}:{}&0.276393202250&{}:{}&0.447213595500&,\\ A^\prime_{\mathbf{6}}&{}\approx{}&0.453848561685&{}:{}&0.208611286432&{}:{}&0.337540151884&,\\B^\prime_{\mathbf{6}}&{}\approx{}&0.208611286432&{}:{}&0.453848561685&{}:{}&0.337540151884&,\\C^\prime_{\mathbf{6}}&{}\approx{}&2.584993206903&{}:{}&2.584993206903&{}:{}&-4.169986413806&. \end{alignedat} \]
6 (220)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6}}&{}\approx{}&0.499615939044&{}:{}&0.499615939044&{}:{}&0.000768121912&,\\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.386066645866&{}:{}&0.612990926709&{}:{}&0.000942427425&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.612990926709&{}:{}&0.386066645866&{}:{}&0.000942427425&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.331229924058&{}:{}&0.331229924058&{}:{}&0.337540151884&. \end{alignedat} \]
6 (220)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6}}&{}\approx{}&0.794586950088&{}:{}&0.794586950088&{}:{}&-0.589173900176&,\\ A^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&0.640033295438&{}:{}&1.392437559508&{}:{}&-1.032470854946&,\\B^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&1.392437559508&{}:{}&0.640033295438&{}:{}&-1.032470854946&,\\C^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&0.364467536191&{}:{}&0.364467536191&{}:{}&0.271064927618&. \end{alignedat} \]
6 (220)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6}}&{}\approx{}&0.418643128010&{}:{}&0.418643128010&{}:{}&0.162713743979&,\\ A^\prime_{\mathbf{6}}&{}\approx{}&0.453848561685&{}:{}&0.208611286432&{}:{}&0.337540151884&,\\B^\prime_{\mathbf{6}}&{}\approx{}&0.208611286432&{}:{}&0.453848561685&{}:{}&0.337540151884&,\\C^\prime_{\mathbf{6}}&{}\approx{}&2.584993206903&{}:{}&2.584993206903&{}:{}&-4.169986413806&,\\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.386066645866&{}:{}&0.612990926709&{}:{}&0.000942427425&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.612990926709&{}:{}&0.386066645866&{}:{}&0.000942427425&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.331229924058&{}:{}&0.331229924058&{}:{}&0.337540151884&, \end{alignedat} \]
6 (220)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6}}&{}\approx{}&0.521023534915&{}:{}&0.521023534915&{}:{}&-0.042047069830&,\\ A^*_{\mathbf{6}}&{}\approx{}&0.000000000000&{}:{}&1.087785252292&{}:{}&-0.087785252292&,\\B^*_{\mathbf{6}}&{}\approx{}&1.087785252292&{}:{}&0.000000000000&{}:{}&-0.087785252292&,\\C^*_{\mathbf{6}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
6 (220)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6}}&{}\approx{}&0.487834409624&{}:{}&0.487834409624&{}:{}&0.024331180752&,\\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.386066645866&{}:{}&0.612990926709&{}:{}&0.000942427425&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.612990926709&{}:{}&0.386066645866&{}:{}&0.000942427425&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.331229924058&{}:{}&0.331229924058&{}:{}&0.337540151884&,\\ A^*_{\mathbf{6}}&{}\approx{}&0.000000000000&{}:{}&1.087785252292&{}:{}&-0.087785252292&,\\B^*_{\mathbf{6}}&{}\approx{}&1.087785252292&{}:{}&0.000000000000&{}:{}&-0.087785252292&,\\C^*_{\mathbf{6}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
6 (220)