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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3(022)

Malfatti circles

3 (022)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.276393202250&{}:{}&0.276393202250&{}:{}&0.447213595500&, \\ P_{\mathbf{3}}&{}\approx{}&0.000614876449&{}:{}&0.977097274982&{}:{}&0.022287848569&, \\ P^-_{\mathbf{3}}&{}\approx{}&-0.032625567140&{}:{}&1.061555377824&{}:{}&-0.028929810685&, \\ P^+_{\mathbf{3}}&{}\approx{}&0.027398651828&{}:{}&0.909044426967&{}:{}&0.063556921204&, \\ Q_{\mathbf{3}}&{}\approx{}&-0.019510332317&{}:{}&0.777749316571&{}:{}&0.241761015746&, \\ I^\prime_{\mathbf{3}}&{}\approx{}&0.020609298995&{}:{}&0.821557928802&{}:{}&0.157832772203&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{3}}\)
\(P^-_{\mathbf{3}}\)
\(P^+_{\mathbf{3}}\)
\(Q_{\mathbf{3}}\)
\(I^\prime_{\mathbf{3}}\)
3 (022)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3}}&{}\approx{}&-0.597613662554&{}:{}&0.610234118204&{}:{}&0.987379544349&,\\B^\prime_{\mathbf{3}}&{}\approx{}&0.029000095800&{}:{}&0.924076763519&{}:{}&0.046923140681&,\\C^\prime_{\mathbf{3}}&{}\approx{}&0.883691590199&{}:{}&0.883691590199&{}:{}&-0.767383180398&, \\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.056157278885&{}:{}&0.922793554837&{}:{}&0.021049166278&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.007465291253&{}:{}&0.721935174254&{}:{}&0.270599534493&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.000572551367&{}:{}&0.909838685235&{}:{}&0.089588763398&, \\ A^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&0.031253684772&{}:{}&0.995886498846&{}:{}&-0.027140183618&,\\B^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&1.155262692143&{}:{}&-1.179659626960&{}:{}&1.024396934817&,\\C^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&-0.030129798552&{}:{}&0.980349231892&{}:{}&0.049780566660&, \\ A^*_{\mathbf{3}}&{}\approx{}&0.000000000000&{}:{}&0.762865556060&{}:{}&0.237134443940&,\\B^*_{\mathbf{3}}&{}\approx{}&-0.087785252292&{}:{}&0.000000000000&{}:{}&1.087785252292&,\\C^*_{\mathbf{3}}&{}\approx{}&-0.025731112119&{}:{}&1.025731112119&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3}}}{B^\prime_{\mathbf{3}}}{C^\prime_{\mathbf{3}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3}}}{B^{\prime\prime}_{\mathbf{3}}}{C^{\prime\prime}_{\mathbf{3}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3}}}{B^{\prime\prime\prime}_{\mathbf{3}}}{C^{\prime\prime\prime}_{\mathbf{3}}}\)
\(\triangle{A^*_{\mathbf{3}}}{B^*_{\mathbf{3}}}{C^*_{\mathbf{3}}}\)
3 (022)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3}}}}&{}\approx{}&2.207847780758&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3}}}}&{}\approx{}&0.104923332281&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3}}}}&{}\approx{}&3.197226208912&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.276393202250&{}:{}&0.276393202250&{}:{}&0.447213595500&,\\ A^\prime_{\mathbf{3}}&{}\approx{}&-0.597613662554&{}:{}&0.610234118204&{}:{}&0.987379544349&,\\B^\prime_{\mathbf{3}}&{}\approx{}&0.029000095800&{}:{}&0.924076763519&{}:{}&0.046923140681&,\\C^\prime_{\mathbf{3}}&{}\approx{}&0.883691590199&{}:{}&0.883691590199&{}:{}&-0.767383180398&. \end{alignedat} \]
3 (022)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3}}&{}\approx{}&0.000614876449&{}:{}&0.977097274982&{}:{}&0.022287848569&,\\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.056157278885&{}:{}&0.922793554837&{}:{}&0.021049166278&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.007465291253&{}:{}&0.721935174254&{}:{}&0.270599534493&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.000572551367&{}:{}&0.909838685235&{}:{}&0.089588763398&. \end{alignedat} \]
3 (022)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3}}&{}\approx{}&-0.032625567140&{}:{}&1.061555377824&{}:{}&-0.028929810685&,\\ A^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&0.031253684772&{}:{}&0.995886498846&{}:{}&-0.027140183618&,\\B^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&1.155262692143&{}:{}&-1.179659626960&{}:{}&1.024396934817&,\\C^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&-0.030129798552&{}:{}&0.980349231892&{}:{}&0.049780566660&. \end{alignedat} \]
3 (022)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3}}&{}\approx{}&0.027398651828&{}:{}&0.909044426967&{}:{}&0.063556921204&,\\ A^\prime_{\mathbf{3}}&{}\approx{}&-0.597613662554&{}:{}&0.610234118204&{}:{}&0.987379544349&,\\B^\prime_{\mathbf{3}}&{}\approx{}&0.029000095800&{}:{}&0.924076763519&{}:{}&0.046923140681&,\\C^\prime_{\mathbf{3}}&{}\approx{}&0.883691590199&{}:{}&0.883691590199&{}:{}&-0.767383180398&,\\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.056157278885&{}:{}&0.922793554837&{}:{}&0.021049166278&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.007465291253&{}:{}&0.721935174254&{}:{}&0.270599534493&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.000572551367&{}:{}&0.909838685235&{}:{}&0.089588763398&, \end{alignedat} \]
3 (022)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3}}&{}\approx{}&-0.019510332317&{}:{}&0.777749316571&{}:{}&0.241761015746&,\\ A^*_{\mathbf{3}}&{}\approx{}&0.000000000000&{}:{}&0.762865556060&{}:{}&0.237134443940&,\\B^*_{\mathbf{3}}&{}\approx{}&-0.087785252292&{}:{}&0.000000000000&{}:{}&1.087785252292&,\\C^*_{\mathbf{3}}&{}\approx{}&-0.025731112119&{}:{}&1.025731112119&{}:{}&0.000000000000&. \end{alignedat} \]
3 (022)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3}}&{}\approx{}&0.020609298995&{}:{}&0.821557928802&{}:{}&0.157832772203&,\\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.056157278885&{}:{}&0.922793554837&{}:{}&0.021049166278&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.007465291253&{}:{}&0.721935174254&{}:{}&0.270599534493&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.000572551367&{}:{}&0.909838685235&{}:{}&0.089588763398&,\\ A^*_{\mathbf{3}}&{}\approx{}&0.000000000000&{}:{}&0.762865556060&{}:{}&0.237134443940&,\\B^*_{\mathbf{3}}&{}\approx{}&-0.087785252292&{}:{}&0.000000000000&{}:{}&1.087785252292&,\\C^*_{\mathbf{3}}&{}\approx{}&-0.025731112119&{}:{}&1.025731112119&{}:{}&0.000000000000&. \end{alignedat} \]
3 (022)