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”.

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


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7b(323)

Malfatti circles

7b (323)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.855555555556&{}:{}&-0.925925925926&{}:{}&1.070370370370&, \\ P_{\mathbf{7b}}&{}\approx{}&-0.006391326057&{}:{}&1.012720831253&{}:{}&-0.006329505195&, \\ P^-_{\mathbf{7b}}&{}\approx{}&0.017287547693&{}:{}&0.959463521794&{}:{}&0.023248930512&, \\ P^+_{\mathbf{7b}}&{}\approx{}&-0.031446817882&{}:{}&1.069074359539&{}:{}&-0.037627541657&, \\ Q_{\mathbf{7b}}&{}\approx{}&0.123893805310&{}:{}&0.778761061947&{}:{}&0.097345132743&, \\ I^\prime_{\mathbf{7b}}&{}\approx{}&-0.091056910569&{}:{}&1.192411924119&{}:{}&-0.101355013550&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{7b}}\)
\(P^-_{\mathbf{7b}}\)
\(P^+_{\mathbf{7b}}\)
\(Q_{\mathbf{7b}}\)
\(I^\prime_{\mathbf{7b}}\)
7b (323)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7b}}&{}\approx{}&1.297916666667&{}:{}&1.909722222222&{}:{}&-2.207638888889&,\\B^\prime_{\mathbf{7b}}&{}\approx{}&-0.025925925926&{}:{}&1.058361391695&{}:{}&-0.032435465769&,\\C^\prime_{\mathbf{7b}}&{}\approx{}&-0.933333333333&{}:{}&1.010101010101&{}:{}&0.923232323232&, \\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.050450450450&{}:{}&1.057057057057&{}:{}&-0.006606606607&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.161441441441&{}:{}&1.321321321321&{}:{}&-0.159879879880&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.006757164404&{}:{}&1.070688788336&{}:{}&-0.063931623932&, \\ A^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.024502784407&{}:{}&1.000265181649&{}:{}&0.024237602758&,\\B^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.262039957939&{}:{}&0.385559060638&{}:{}&0.352400981423&,\\C^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.018248388987&{}:{}&1.012790470611&{}:{}&-0.031038859598&, \\ A^*_{\mathbf{7b}}&{}\approx{}&0.000000000000&{}:{}&0.888888888889&{}:{}&0.111111111111&,\\B^*_{\mathbf{7b}}&{}\approx{}&0.560000000000&{}:{}&0.000000000000&{}:{}&0.440000000000&,\\C^*_{\mathbf{7b}}&{}\approx{}&0.137254901961&{}:{}&0.862745098039&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7b}}}{B^\prime_{\mathbf{7b}}}{C^\prime_{\mathbf{7b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7b}}}{B^{\prime\prime}_{\mathbf{7b}}}{C^{\prime\prime}_{\mathbf{7b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7b}}}{B^{\prime\prime\prime}_{\mathbf{7b}}}{C^{\prime\prime\prime}_{\mathbf{7b}}}\)
\(\triangle{A^*_{\mathbf{7b}}}{B^*_{\mathbf{7b}}}{C^*_{\mathbf{7b}}}\)
7b (323)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7b}}}}&{}\approx{}&-2.062500000000&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7b}}}}&{}\approx{}&-0.030303030303&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7b}}}}&{}\approx{}&-1.090909090909&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.855555555556&{}:{}&-0.925925925926&{}:{}&1.070370370370&,\\ A^\prime_{\mathbf{7b}}&{}\approx{}&1.297916666667&{}:{}&1.909722222222&{}:{}&-2.207638888889&,\\B^\prime_{\mathbf{7b}}&{}\approx{}&-0.025925925926&{}:{}&1.058361391695&{}:{}&-0.032435465769&,\\C^\prime_{\mathbf{7b}}&{}\approx{}&-0.933333333333&{}:{}&1.010101010101&{}:{}&0.923232323232&. \end{alignedat} \]
7b (323)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7b}}&{}\approx{}&-0.006391326057&{}:{}&1.012720831253&{}:{}&-0.006329505195&,\\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.050450450450&{}:{}&1.057057057057&{}:{}&-0.006606606607&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.161441441441&{}:{}&1.321321321321&{}:{}&-0.159879879880&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.006757164404&{}:{}&1.070688788336&{}:{}&-0.063931623932&. \end{alignedat} \]
7b (323)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7b}}&{}\approx{}&0.017287547693&{}:{}&0.959463521794&{}:{}&0.023248930512&,\\ A^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.024502784407&{}:{}&1.000265181649&{}:{}&0.024237602758&,\\B^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.262039957939&{}:{}&0.385559060638&{}:{}&0.352400981423&,\\C^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.018248388987&{}:{}&1.012790470611&{}:{}&-0.031038859598&. \end{alignedat} \]
7b (323)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7b}}&{}\approx{}&-0.031446817882&{}:{}&1.069074359539&{}:{}&-0.037627541657&,\\ A^\prime_{\mathbf{7b}}&{}\approx{}&1.297916666667&{}:{}&1.909722222222&{}:{}&-2.207638888889&,\\B^\prime_{\mathbf{7b}}&{}\approx{}&-0.025925925926&{}:{}&1.058361391695&{}:{}&-0.032435465769&,\\C^\prime_{\mathbf{7b}}&{}\approx{}&-0.933333333333&{}:{}&1.010101010101&{}:{}&0.923232323232&,\\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.050450450450&{}:{}&1.057057057057&{}:{}&-0.006606606607&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.161441441441&{}:{}&1.321321321321&{}:{}&-0.159879879880&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.006757164404&{}:{}&1.070688788336&{}:{}&-0.063931623932&, \end{alignedat} \]
7b (323)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7b}}&{}\approx{}&0.123893805310&{}:{}&0.778761061947&{}:{}&0.097345132743&,\\ A^*_{\mathbf{7b}}&{}\approx{}&0.000000000000&{}:{}&0.888888888889&{}:{}&0.111111111111&,\\B^*_{\mathbf{7b}}&{}\approx{}&0.560000000000&{}:{}&0.000000000000&{}:{}&0.440000000000&,\\C^*_{\mathbf{7b}}&{}\approx{}&0.137254901961&{}:{}&0.862745098039&{}:{}&0.000000000000&. \end{alignedat} \]
7b (323)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7b}}&{}\approx{}&-0.091056910569&{}:{}&1.192411924119&{}:{}&-0.101355013550&,\\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.050450450450&{}:{}&1.057057057057&{}:{}&-0.006606606607&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.161441441441&{}:{}&1.321321321321&{}:{}&-0.159879879880&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.006757164404&{}:{}&1.070688788336&{}:{}&-0.063931623932&,\\ A^*_{\mathbf{7b}}&{}\approx{}&0.000000000000&{}:{}&0.888888888889&{}:{}&0.111111111111&,\\B^*_{\mathbf{7b}}&{}\approx{}&0.560000000000&{}:{}&0.000000000000&{}:{}&0.440000000000&,\\C^*_{\mathbf{7b}}&{}\approx{}&0.137254901961&{}:{}&0.862745098039&{}:{}&0.000000000000&. \end{alignedat} \]
7b (323)