Derousseau's Generalization of the Malfatti circles

Isosceles Triangle with 135° Top Angle

\(A=22.5\degree\), \(B=22.5\degree\), \(C=135\degree\).


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6(220)

Malfatti circles

6 (220)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.259891532474&{}:{}&0.259891532474&{}:{}&0.480216935052&, \\ P_{\mathbf{6}}&{}\approx{}&0.499910807065&{}:{}&0.499910807065&{}:{}&0.000178385870&, \\ P^-_{\mathbf{6}}&{}\approx{}&1.048341617171&{}:{}&1.048341617171&{}:{}&-1.096683234343&, \\ P^+_{\mathbf{6}}&{}\approx{}&0.401447312137&{}:{}&0.401447312137&{}:{}&0.197105375726&, \\ Q_{\mathbf{6}}&{}\approx{}&0.517012249752&{}:{}&0.517012249752&{}:{}&-0.034024499505&, \\ I^\prime_{\mathbf{6}}&{}\approx{}&0.493661940902&{}:{}&0.493661940902&{}:{}&0.012676118196&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{6}}\)
\(P^-_{\mathbf{6}}\)
\(P^+_{\mathbf{6}}\)
\(Q_{\mathbf{6}}\)
\(I^\prime_{\mathbf{6}}\)
6 (220)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6}}&{}\approx{}&0.382683432365&{}:{}&0.216772751325&{}:{}&0.400543816310&,\\B^\prime_{\mathbf{6}}&{}\approx{}&0.216772751325&{}:{}&0.382683432365&{}:{}&0.400543816310&,\\C^\prime_{\mathbf{6}}&{}\approx{}&6.526234467956&{}:{}&6.526234467956&{}:{}&-12.052468935912&, \\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.419607974722&{}:{}&0.580184994736&{}:{}&0.000207030542&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.580184994736&{}:{}&0.419607974722&{}:{}&0.000207030542&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.299728091845&{}:{}&0.299728091845&{}:{}&0.400543816310&, \\ A^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&1.088619529491&{}:{}&1.921812845662&{}:{}&-2.010432375154&,\\B^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&1.921812845662&{}:{}&1.088619529491&{}:{}&-2.010432375154&,\\C^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&0.328227057688&{}:{}&0.328227057688&{}:{}&0.343545884624&, \\ A^*_{\mathbf{6}}&{}\approx{}&0.000000000000&{}:{}&1.070445884989&{}:{}&-0.070445884989&,\\B^*_{\mathbf{6}}&{}\approx{}&1.070445884989&{}:{}&0.000000000000&{}:{}&-0.070445884989&,\\C^*_{\mathbf{6}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6}}}{B^\prime_{\mathbf{6}}}{C^\prime_{\mathbf{6}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6}}}{B^{\prime\prime}_{\mathbf{6}}}{C^{\prime\prime}_{\mathbf{6}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6}}}{B^{\prime\prime\prime}_{\mathbf{6}}}{C^{\prime\prime\prime}_{\mathbf{6}}}\)
\(\triangle{A^*_{\mathbf{6}}}{B^*_{\mathbf{6}}}{C^*_{\mathbf{6}}}\)
6 (220)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6}}}}&{}\approx{}&0.834089318960&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6}}}}&{}\approx{}&0.834089318960&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6}}}}&{}\approx{}&25.111377834540&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.259891532474&{}:{}&0.259891532474&{}:{}&0.480216935052&,\\ A^\prime_{\mathbf{6}}&{}\approx{}&0.382683432365&{}:{}&0.216772751325&{}:{}&0.400543816310&,\\B^\prime_{\mathbf{6}}&{}\approx{}&0.216772751325&{}:{}&0.382683432365&{}:{}&0.400543816310&,\\C^\prime_{\mathbf{6}}&{}\approx{}&6.526234467956&{}:{}&6.526234467956&{}:{}&-12.052468935912&. \end{alignedat} \]
6 (220)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6}}&{}\approx{}&0.499910807065&{}:{}&0.499910807065&{}:{}&0.000178385870&,\\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.419607974722&{}:{}&0.580184994736&{}:{}&0.000207030542&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.580184994736&{}:{}&0.419607974722&{}:{}&0.000207030542&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.299728091845&{}:{}&0.299728091845&{}:{}&0.400543816310&. \end{alignedat} \]
6 (220)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6}}&{}\approx{}&1.048341617171&{}:{}&1.048341617171&{}:{}&-1.096683234343&,\\ A^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&1.088619529491&{}:{}&1.921812845662&{}:{}&-2.010432375154&,\\B^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&1.921812845662&{}:{}&1.088619529491&{}:{}&-2.010432375154&,\\C^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&0.328227057688&{}:{}&0.328227057688&{}:{}&0.343545884624&. \end{alignedat} \]
6 (220)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6}}&{}\approx{}&0.401447312137&{}:{}&0.401447312137&{}:{}&0.197105375726&,\\ A^\prime_{\mathbf{6}}&{}\approx{}&0.382683432365&{}:{}&0.216772751325&{}:{}&0.400543816310&,\\B^\prime_{\mathbf{6}}&{}\approx{}&0.216772751325&{}:{}&0.382683432365&{}:{}&0.400543816310&,\\C^\prime_{\mathbf{6}}&{}\approx{}&6.526234467956&{}:{}&6.526234467956&{}:{}&-12.052468935912&,\\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.419607974722&{}:{}&0.580184994736&{}:{}&0.000207030542&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.580184994736&{}:{}&0.419607974722&{}:{}&0.000207030542&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.299728091845&{}:{}&0.299728091845&{}:{}&0.400543816310&, \end{alignedat} \]
6 (220)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6}}&{}\approx{}&0.517012249752&{}:{}&0.517012249752&{}:{}&-0.034024499505&,\\ A^*_{\mathbf{6}}&{}\approx{}&0.000000000000&{}:{}&1.070445884989&{}:{}&-0.070445884989&,\\B^*_{\mathbf{6}}&{}\approx{}&1.070445884989&{}:{}&0.000000000000&{}:{}&-0.070445884989&,\\C^*_{\mathbf{6}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
6 (220)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6}}&{}\approx{}&0.493661940902&{}:{}&0.493661940902&{}:{}&0.012676118196&,\\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.419607974722&{}:{}&0.580184994736&{}:{}&0.000207030542&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.580184994736&{}:{}&0.419607974722&{}:{}&0.000207030542&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.299728091845&{}:{}&0.299728091845&{}:{}&0.400543816310&,\\ A^*_{\mathbf{6}}&{}\approx{}&0.000000000000&{}:{}&1.070445884989&{}:{}&-0.070445884989&,\\B^*_{\mathbf{6}}&{}\approx{}&1.070445884989&{}:{}&0.000000000000&{}:{}&-0.070445884989&,\\C^*_{\mathbf{6}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
6 (220)