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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5(202)

Malfatti circles

5 (202)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.276393202250&{}:{}&0.276393202250&{}:{}&0.447213595500&, \\ P_{\mathbf{5}}&{}\approx{}&0.977097274982&{}:{}&0.000614876449&{}:{}&0.022287848569&, \\ P^-_{\mathbf{5}}&{}\approx{}&1.061555377824&{}:{}&-0.032625567140&{}:{}&-0.028929810685&, \\ P^+_{\mathbf{5}}&{}\approx{}&0.909044426967&{}:{}&0.027398651828&{}:{}&0.063556921204&, \\ Q_{\mathbf{5}}&{}\approx{}&0.777749316571&{}:{}&-0.019510332317&{}:{}&0.241761015746&, \\ I^\prime_{\mathbf{5}}&{}\approx{}&0.821557928802&{}:{}&0.020609298995&{}:{}&0.157832772203&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{5}}\)
\(P^-_{\mathbf{5}}\)
\(P^+_{\mathbf{5}}\)
\(Q_{\mathbf{5}}\)
\(I^\prime_{\mathbf{5}}\)
5 (202)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5}}&{}\approx{}&0.924076763519&{}:{}&0.029000095800&{}:{}&0.046923140681&,\\B^\prime_{\mathbf{5}}&{}\approx{}&0.610234118204&{}:{}&-0.597613662554&{}:{}&0.987379544349&,\\C^\prime_{\mathbf{5}}&{}\approx{}&0.883691590199&{}:{}&0.883691590199&{}:{}&-0.767383180398&, \\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.721935174254&{}:{}&0.007465291253&{}:{}&0.270599534493&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.922793554837&{}:{}&0.056157278885&{}:{}&0.021049166278&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.909838685235&{}:{}&0.000572551367&{}:{}&0.089588763398&, \\ A^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&-1.179659626960&{}:{}&1.155262692143&{}:{}&1.024396934817&,\\B^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&0.995886498846&{}:{}&0.031253684772&{}:{}&-0.027140183618&,\\C^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&0.980349231892&{}:{}&-0.030129798552&{}:{}&0.049780566660&, \\ A^*_{\mathbf{5}}&{}\approx{}&0.000000000000&{}:{}&-0.087785252292&{}:{}&1.087785252292&,\\B^*_{\mathbf{5}}&{}\approx{}&0.762865556060&{}:{}&0.000000000000&{}:{}&0.237134443940&,\\C^*_{\mathbf{5}}&{}\approx{}&1.025731112119&{}:{}&-0.025731112119&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5}}}{B^\prime_{\mathbf{5}}}{C^\prime_{\mathbf{5}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5}}}{B^{\prime\prime}_{\mathbf{5}}}{C^{\prime\prime}_{\mathbf{5}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5}}}{B^{\prime\prime\prime}_{\mathbf{5}}}{C^{\prime\prime\prime}_{\mathbf{5}}}\)
\(\triangle{A^*_{\mathbf{5}}}{B^*_{\mathbf{5}}}{C^*_{\mathbf{5}}}\)
5 (202)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5}}}}&{}\approx{}&0.104923332281&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5}}}}&{}\approx{}&2.207847780758&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5}}}}&{}\approx{}&3.197226208912&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.276393202250&{}:{}&0.276393202250&{}:{}&0.447213595500&,\\ A^\prime_{\mathbf{5}}&{}\approx{}&0.924076763519&{}:{}&0.029000095800&{}:{}&0.046923140681&,\\B^\prime_{\mathbf{5}}&{}\approx{}&0.610234118204&{}:{}&-0.597613662554&{}:{}&0.987379544349&,\\C^\prime_{\mathbf{5}}&{}\approx{}&0.883691590199&{}:{}&0.883691590199&{}:{}&-0.767383180398&. \end{alignedat} \]
5 (202)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5}}&{}\approx{}&0.977097274982&{}:{}&0.000614876449&{}:{}&0.022287848569&,\\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.721935174254&{}:{}&0.007465291253&{}:{}&0.270599534493&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.922793554837&{}:{}&0.056157278885&{}:{}&0.021049166278&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.909838685235&{}:{}&0.000572551367&{}:{}&0.089588763398&. \end{alignedat} \]
5 (202)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5}}&{}\approx{}&1.061555377824&{}:{}&-0.032625567140&{}:{}&-0.028929810685&,\\ A^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&-1.179659626960&{}:{}&1.155262692143&{}:{}&1.024396934817&,\\B^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&0.995886498846&{}:{}&0.031253684772&{}:{}&-0.027140183618&,\\C^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&0.980349231892&{}:{}&-0.030129798552&{}:{}&0.049780566660&. \end{alignedat} \]
5 (202)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5}}&{}\approx{}&0.909044426967&{}:{}&0.027398651828&{}:{}&0.063556921204&,\\ A^\prime_{\mathbf{5}}&{}\approx{}&0.924076763519&{}:{}&0.029000095800&{}:{}&0.046923140681&,\\B^\prime_{\mathbf{5}}&{}\approx{}&0.610234118204&{}:{}&-0.597613662554&{}:{}&0.987379544349&,\\C^\prime_{\mathbf{5}}&{}\approx{}&0.883691590199&{}:{}&0.883691590199&{}:{}&-0.767383180398&,\\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.721935174254&{}:{}&0.007465291253&{}:{}&0.270599534493&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.922793554837&{}:{}&0.056157278885&{}:{}&0.021049166278&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.909838685235&{}:{}&0.000572551367&{}:{}&0.089588763398&, \end{alignedat} \]
5 (202)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5}}&{}\approx{}&0.777749316571&{}:{}&-0.019510332317&{}:{}&0.241761015746&,\\ A^*_{\mathbf{5}}&{}\approx{}&0.000000000000&{}:{}&-0.087785252292&{}:{}&1.087785252292&,\\B^*_{\mathbf{5}}&{}\approx{}&0.762865556060&{}:{}&0.000000000000&{}:{}&0.237134443940&,\\C^*_{\mathbf{5}}&{}\approx{}&1.025731112119&{}:{}&-0.025731112119&{}:{}&0.000000000000&. \end{alignedat} \]
5 (202)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5}}&{}\approx{}&0.821557928802&{}:{}&0.020609298995&{}:{}&0.157832772203&,\\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.721935174254&{}:{}&0.007465291253&{}:{}&0.270599534493&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.922793554837&{}:{}&0.056157278885&{}:{}&0.021049166278&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.909838685235&{}:{}&0.000572551367&{}:{}&0.089588763398&,\\ A^*_{\mathbf{5}}&{}\approx{}&0.000000000000&{}:{}&-0.087785252292&{}:{}&1.087785252292&,\\B^*_{\mathbf{5}}&{}\approx{}&0.762865556060&{}:{}&0.000000000000&{}:{}&0.237134443940&,\\C^*_{\mathbf{5}}&{}\approx{}&1.025731112119&{}:{}&-0.025731112119&{}:{}&0.000000000000&. \end{alignedat} \]
5 (202)