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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4(200)

Malfatti circles

4 (200)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.300000000000&{}:{}&0.324675324675&{}:{}&0.375324675325&, \\ P_{\mathbf{4}}&{}\approx{}&0.993911224727&{}:{}&0.002689153747&{}:{}&0.003399621527&, \\ P^-_{\mathbf{4}}&{}\approx{}&1.027381242387&{}:{}&-0.012841482513&{}:{}&-0.014539759875&, \\ P^+_{\mathbf{4}}&{}\approx{}&0.963385914241&{}:{}&0.016853397658&{}:{}&0.019760688101&, \\ Q_{\mathbf{4}}&{}\approx{}&1.148936170213&{}:{}&-0.063829787234&{}:{}&-0.085106382979&, \\ I^\prime_{\mathbf{4}}&{}\approx{}&0.893233082707&{}:{}&0.048335123523&{}:{}&0.058431793770&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{4}}\)
\(P^-_{\mathbf{4}}\)
\(P^+_{\mathbf{4}}\)
\(Q_{\mathbf{4}}\)
\(I^\prime_{\mathbf{4}}\)
4 (200)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4}}&{}\approx{}&0.968181818182&{}:{}&0.014757969303&{}:{}&0.017060212515&,\\B^\prime_{\mathbf{4}}&{}\approx{}&0.733333333333&{}:{}&-0.650793650794&{}:{}&0.917460317460&,\\C^\prime_{\mathbf{4}}&{}\approx{}&0.928125000000&{}:{}&1.004464285714&{}:{}&-0.932589285714&, \\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.819310344828&{}:{}&0.079802955665&{}:{}&0.100886699507&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.967601809955&{}:{}&0.029088558500&{}:{}&0.003309631545&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.963894523327&{}:{}&0.002607939728&{}:{}&0.033497536946&, \\ A^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&-1.120754716981&{}:{}&0.994609164420&{}:{}&1.126145552561&,\\B^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&0.998910468972&{}:{}&0.015226365297&{}:{}&-0.014136834269&,\\C^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&0.994904458599&{}:{}&-0.012435547467&{}:{}&0.017531088869&, \\ A^*_{\mathbf{4}}&{}\approx{}&0.000000000000&{}:{}&0.428571428571&{}:{}&0.571428571429&,\\B^*_{\mathbf{4}}&{}\approx{}&1.080000000000&{}:{}&0.000000000000&{}:{}&-0.080000000000&,\\C^*_{\mathbf{4}}&{}\approx{}&1.058823529412&{}:{}&-0.058823529412&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4}}}{B^\prime_{\mathbf{4}}}{C^\prime_{\mathbf{4}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4}}}{B^{\prime\prime}_{\mathbf{4}}}{C^{\prime\prime}_{\mathbf{4}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4}}}{B^{\prime\prime\prime}_{\mathbf{4}}}{C^{\prime\prime\prime}_{\mathbf{4}}}\)
\(\triangle{A^*_{\mathbf{4}}}{B^*_{\mathbf{4}}}{C^*_{\mathbf{4}}}\)
4 (200)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4}}}}&{}\approx{}&0.045454545455&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4}}}}&{}\approx{}&2.444444444444&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4}}}}&{}\approx{}&3.093750000000&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.300000000000&{}:{}&0.324675324675&{}:{}&0.375324675325&,\\ A^\prime_{\mathbf{4}}&{}\approx{}&0.968181818182&{}:{}&0.014757969303&{}:{}&0.017060212515&,\\B^\prime_{\mathbf{4}}&{}\approx{}&0.733333333333&{}:{}&-0.650793650794&{}:{}&0.917460317460&,\\C^\prime_{\mathbf{4}}&{}\approx{}&0.928125000000&{}:{}&1.004464285714&{}:{}&-0.932589285714&. \end{alignedat} \]
4 (200)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4}}&{}\approx{}&0.993911224727&{}:{}&0.002689153747&{}:{}&0.003399621527&,\\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.819310344828&{}:{}&0.079802955665&{}:{}&0.100886699507&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.967601809955&{}:{}&0.029088558500&{}:{}&0.003309631545&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.963894523327&{}:{}&0.002607939728&{}:{}&0.033497536946&. \end{alignedat} \]
4 (200)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4}}&{}\approx{}&1.027381242387&{}:{}&-0.012841482513&{}:{}&-0.014539759875&,\\ A^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&-1.120754716981&{}:{}&0.994609164420&{}:{}&1.126145552561&,\\B^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&0.998910468972&{}:{}&0.015226365297&{}:{}&-0.014136834269&,\\C^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&0.994904458599&{}:{}&-0.012435547467&{}:{}&0.017531088869&. \end{alignedat} \]
4 (200)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4}}&{}\approx{}&0.963385914241&{}:{}&0.016853397658&{}:{}&0.019760688101&,\\ A^\prime_{\mathbf{4}}&{}\approx{}&0.968181818182&{}:{}&0.014757969303&{}:{}&0.017060212515&,\\B^\prime_{\mathbf{4}}&{}\approx{}&0.733333333333&{}:{}&-0.650793650794&{}:{}&0.917460317460&,\\C^\prime_{\mathbf{4}}&{}\approx{}&0.928125000000&{}:{}&1.004464285714&{}:{}&-0.932589285714&,\\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.819310344828&{}:{}&0.079802955665&{}:{}&0.100886699507&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.967601809955&{}:{}&0.029088558500&{}:{}&0.003309631545&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.963894523327&{}:{}&0.002607939728&{}:{}&0.033497536946&, \end{alignedat} \]
4 (200)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4}}&{}\approx{}&1.148936170213&{}:{}&-0.063829787234&{}:{}&-0.085106382979&,\\ A^*_{\mathbf{4}}&{}\approx{}&0.000000000000&{}:{}&0.428571428571&{}:{}&0.571428571429&,\\B^*_{\mathbf{4}}&{}\approx{}&1.080000000000&{}:{}&0.000000000000&{}:{}&-0.080000000000&,\\C^*_{\mathbf{4}}&{}\approx{}&1.058823529412&{}:{}&-0.058823529412&{}:{}&0.000000000000&. \end{alignedat} \]
4 (200)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4}}&{}\approx{}&0.893233082707&{}:{}&0.048335123523&{}:{}&0.058431793770&,\\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.819310344828&{}:{}&0.079802955665&{}:{}&0.100886699507&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.967601809955&{}:{}&0.029088558500&{}:{}&0.003309631545&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.963894523327&{}:{}&0.002607939728&{}:{}&0.033497536946&,\\ A^*_{\mathbf{4}}&{}\approx{}&0.000000000000&{}:{}&0.428571428571&{}:{}&0.571428571429&,\\B^*_{\mathbf{4}}&{}\approx{}&1.080000000000&{}:{}&0.000000000000&{}:{}&-0.080000000000&,\\C^*_{\mathbf{4}}&{}\approx{}&1.058823529412&{}:{}&-0.058823529412&{}:{}&0.000000000000&. \end{alignedat} \]
4 (200)