Derousseau's Generalization of the Malfatti circles

Martin's solution

Problem 4331 (proposed by A. Martin) I. Solution by the Proposer, Mathematical Questions with their Solutions, from the “Educational Times”.

\(a:b:c=231:250:289\).

[Top] > Martin's solution > 5b (303)

5b (303)

Malfatti circles

5b (303)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.855555555556&{}:{}&-0.925925925926&{}:{}&1.070370370370&, \\ P_{\mathbf{5b}}&{}\approx{}&0.285960903783&{}:{}&-0.029025979521&{}:{}&0.743065075738&, \\ P^-_{\mathbf{5b}}&{}\approx{}&0.202387190684&{}:{}&0.102571567200&{}:{}&0.695041242115&, \\ P^+_{\mathbf{5b}}&{}\approx{}&0.350573936529&{}:{}&-0.130767499437&{}:{}&0.780193562908&, \\ Q_{\mathbf{5b}}&{}\approx{}&0.411214953271&{}:{}&0.065420560748&{}:{}&0.523364485981&, \\ I^\prime_{\mathbf{5b}}&{}\approx{}&0.404597701149&{}:{}&-0.134099616858&{}:{}&0.729501915709&, \end{alignedat} \]
\(I_{\mathbf{b}}\) Incenter
\(P_{\mathbf{5b}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{5b}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{5b}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{5b}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{5b}}\) Radical center of the Malfatti circles
5b (303)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5b}}&{}\approx{}&0.905208333333&{}:{}&-0.607638888889&{}:{}&0.702430555556&,\\B^\prime_{\mathbf{5b}}&{}\approx{}&0.325925925926&{}:{}&0.266313932981&{}:{}&0.407760141093&,\\C^\prime_{\mathbf{5b}}&{}\approx{}&0.183333333333&{}:{}&-0.198412698413&{}:{}&1.015079365079&, \\ A^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.234666666667&{}:{}&-0.031111111111&{}:{}&0.796444444444&,\\B^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.361025641026&{}:{}&-0.299145299145&{}:{}&0.938119658120&,\\C^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.538689622190&{}:{}&-0.054678782082&{}:{}&0.515989159892&, \\ A^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.135982339956&{}:{}&0.111111111111&{}:{}&0.752906548933&,\\B^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.265749235474&{}:{}&-0.178389398573&{}:{}&0.912640163099&,\\C^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.438059231254&{}:{}&0.222012182315&{}:{}&0.339928586431&, \\ A^*_{\mathbf{5b}}&{}\approx{}&0.000000000000&{}:{}&0.111111111111&{}:{}&0.888888888889&,\\B^*_{\mathbf{5b}}&{}\approx{}&0.440000000000&{}:{}&0.000000000000&{}:{}&0.560000000000&,\\C^*_{\mathbf{5b}}&{}\approx{}&0.862745098039&{}:{}&0.137254901961&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5b}}}{B^\prime_{\mathbf{5b}}}{C^\prime_{\mathbf{5b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5b}}}{B^{\prime\prime}_{\mathbf{5b}}}{C^{\prime\prime}_{\mathbf{5b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5b}}}{B^{\prime\prime\prime}_{\mathbf{5b}}}{C^{\prime\prime\prime}_{\mathbf{5b}}}\)
\(\triangle{A^*_{\mathbf{5b}}}{B^*_{\mathbf{5b}}}{C^*_{\mathbf{5b}}}\)
5b (303)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5b}}}}&{}\approx{}&0.656250000000&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5b}}}}&{}\approx{}&0.380952380952&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5b}}}}&{}\approx{}&0.214285714286&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.855555555556&{}:{}&-0.925925925926&{}:{}&1.070370370370&,\\ A^\prime_{\mathbf{5b}}&{}\approx{}&0.905208333333&{}:{}&-0.607638888889&{}:{}&0.702430555556&,\\B^\prime_{\mathbf{5b}}&{}\approx{}&0.325925925926&{}:{}&0.266313932981&{}:{}&0.407760141093&,\\C^\prime_{\mathbf{5b}}&{}\approx{}&0.183333333333&{}:{}&-0.198412698413&{}:{}&1.015079365079&. \end{alignedat} \]
5b (303)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5b}}&{}\approx{}&0.285960903783&{}:{}&-0.029025979521&{}:{}&0.743065075738&,\\ A^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.234666666667&{}:{}&-0.031111111111&{}:{}&0.796444444444&,\\B^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.361025641026&{}:{}&-0.299145299145&{}:{}&0.938119658120&,\\C^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.538689622190&{}:{}&-0.054678782082&{}:{}&0.515989159892&. \end{alignedat} \]
5b (303)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5b}}&{}\approx{}&0.202387190684&{}:{}&0.102571567200&{}:{}&0.695041242115&,\\ A^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.135982339956&{}:{}&0.111111111111&{}:{}&0.752906548933&,\\B^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.265749235474&{}:{}&-0.178389398573&{}:{}&0.912640163099&,\\C^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.438059231254&{}:{}&0.222012182315&{}:{}&0.339928586431&. \end{alignedat} \]
5b (303)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5b}}&{}\approx{}&0.350573936529&{}:{}&-0.130767499437&{}:{}&0.780193562908&,\\ A^\prime_{\mathbf{5b}}&{}\approx{}&0.905208333333&{}:{}&-0.607638888889&{}:{}&0.702430555556&,\\B^\prime_{\mathbf{5b}}&{}\approx{}&0.325925925926&{}:{}&0.266313932981&{}:{}&0.407760141093&,\\C^\prime_{\mathbf{5b}}&{}\approx{}&0.183333333333&{}:{}&-0.198412698413&{}:{}&1.015079365079&,\\ A^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.234666666667&{}:{}&-0.031111111111&{}:{}&0.796444444444&,\\B^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.361025641026&{}:{}&-0.299145299145&{}:{}&0.938119658120&,\\C^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.538689622190&{}:{}&-0.054678782082&{}:{}&0.515989159892&, \end{alignedat} \]
5b (303)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5b}}&{}\approx{}&0.411214953271&{}:{}&0.065420560748&{}:{}&0.523364485981&,\\ A^*_{\mathbf{5b}}&{}\approx{}&0.000000000000&{}:{}&0.111111111111&{}:{}&0.888888888889&,\\B^*_{\mathbf{5b}}&{}\approx{}&0.440000000000&{}:{}&0.000000000000&{}:{}&0.560000000000&,\\C^*_{\mathbf{5b}}&{}\approx{}&0.862745098039&{}:{}&0.137254901961&{}:{}&0.000000000000&. \end{alignedat} \]
5b (303)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5b}}&{}\approx{}&0.404597701149&{}:{}&-0.134099616858&{}:{}&0.729501915709&,\\ A^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.234666666667&{}:{}&-0.031111111111&{}:{}&0.796444444444&,\\B^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.361025641026&{}:{}&-0.299145299145&{}:{}&0.938119658120&,\\C^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.538689622190&{}:{}&-0.054678782082&{}:{}&0.515989159892&,\\ A^*_{\mathbf{5b}}&{}\approx{}&0.000000000000&{}:{}&0.111111111111&{}:{}&0.888888888889&,\\B^*_{\mathbf{5b}}&{}\approx{}&0.440000000000&{}:{}&0.000000000000&{}:{}&0.560000000000&,\\C^*_{\mathbf{5b}}&{}\approx{}&0.862745098039&{}:{}&0.137254901961&{}:{}&0.000000000000&. \end{alignedat} \]
5b (303)