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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6 (220)

Malfatti circles

6 (220)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.300000000000&{}:{}&0.324675324675&{}:{}&0.375324675325&, \\ P_{\mathbf{6}}&{}\approx{}&0.589601386482&{}:{}&0.408381912715&{}:{}&0.002016700804&, \\ P^-_{\mathbf{6}}&{}\approx{}&0.814153846154&{}:{}&0.473286713287&{}:{}&-0.287440559441&, \\ P^+_{\mathbf{6}}&{}\approx{}&0.501568154403&{}:{}&0.382936725518&{}:{}&0.115495120079&, \\ Q_{\mathbf{6}}&{}\approx{}&0.551020408163&{}:{}&0.489795918367&{}:{}&-0.040816326531&, \\ I^\prime_{\mathbf{6}}&{}\approx{}&0.517808219178&{}:{}&0.448318804483&{}:{}&0.033872976339&, \end{alignedat} \]
\(I\) Incenter
\(P_{\mathbf{6}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{6}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{6}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{6}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{6}}\) Radical center of the Malfatti circles
6 (220)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6}}&{}\approx{}&0.600000000000&{}:{}&0.185528756957&{}:{}&0.214471243043&,\\B^\prime_{\mathbf{6}}&{}\approx{}&0.233333333333&{}:{}&0.474747474747&{}:{}&0.291919191919&,\\C^\prime_{\mathbf{6}}&{}\approx{}&1.181250000000&{}:{}&1.278409090909&{}:{}&-1.459659090909&, \\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.389690721649&{}:{}&0.607310215558&{}:{}&0.002999062793&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.673663366337&{}:{}&0.324032403240&{}:{}&0.002304230423&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.444705882353&{}:{}&0.308021390374&{}:{}&0.247272727273&, \\ A^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&0.555882352941&{}:{}&1.131016042781&{}:{}&-0.686898395722&,\\B^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&1.045849802372&{}:{}&0.323392022997&{}:{}&-0.369241825368&,\\C^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&0.515789473684&{}:{}&0.299840510367&{}:{}&0.184370015949&, \\ A^*_{\mathbf{6}}&{}\approx{}&0.000000000000&{}:{}&1.090909090909&{}:{}&-0.090909090909&,\\B^*_{\mathbf{6}}&{}\approx{}&1.080000000000&{}:{}&0.000000000000&{}:{}&-0.080000000000&,\\C^*_{\mathbf{6}}&{}\approx{}&0.529411764706&{}:{}&0.470588235294&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6}}}{B^\prime_{\mathbf{6}}}{C^\prime_{\mathbf{6}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6}}}{B^{\prime\prime}_{\mathbf{6}}}{C^{\prime\prime}_{\mathbf{6}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6}}}{B^{\prime\prime\prime}_{\mathbf{6}}}{C^{\prime\prime\prime}_{\mathbf{6}}}\)
\(\triangle{A^*_{\mathbf{6}}}{B^*_{\mathbf{6}}}{C^*_{\mathbf{6}}}\)
6 (220)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6}}}}&{}\approx{}&0.571428571429&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6}}}}&{}\approx{}&0.777777777778&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6}}}}&{}\approx{}&3.937500000000&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.300000000000&{}:{}&0.324675324675&{}:{}&0.375324675325&,\\ A^\prime_{\mathbf{6}}&{}\approx{}&0.600000000000&{}:{}&0.185528756957&{}:{}&0.214471243043&,\\B^\prime_{\mathbf{6}}&{}\approx{}&0.233333333333&{}:{}&0.474747474747&{}:{}&0.291919191919&,\\C^\prime_{\mathbf{6}}&{}\approx{}&1.181250000000&{}:{}&1.278409090909&{}:{}&-1.459659090909&. \end{alignedat} \]
6 (220)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6}}&{}\approx{}&0.589601386482&{}:{}&0.408381912715&{}:{}&0.002016700804&,\\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.389690721649&{}:{}&0.607310215558&{}:{}&0.002999062793&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.673663366337&{}:{}&0.324032403240&{}:{}&0.002304230423&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.444705882353&{}:{}&0.308021390374&{}:{}&0.247272727273&. \end{alignedat} \]
6 (220)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6}}&{}\approx{}&0.814153846154&{}:{}&0.473286713287&{}:{}&-0.287440559441&,\\ A^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&0.555882352941&{}:{}&1.131016042781&{}:{}&-0.686898395722&,\\B^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&1.045849802372&{}:{}&0.323392022997&{}:{}&-0.369241825368&,\\C^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&0.515789473684&{}:{}&0.299840510367&{}:{}&0.184370015949&. \end{alignedat} \]
6 (220)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6}}&{}\approx{}&0.501568154403&{}:{}&0.382936725518&{}:{}&0.115495120079&,\\ A^\prime_{\mathbf{6}}&{}\approx{}&0.600000000000&{}:{}&0.185528756957&{}:{}&0.214471243043&,\\B^\prime_{\mathbf{6}}&{}\approx{}&0.233333333333&{}:{}&0.474747474747&{}:{}&0.291919191919&,\\C^\prime_{\mathbf{6}}&{}\approx{}&1.181250000000&{}:{}&1.278409090909&{}:{}&-1.459659090909&,\\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.389690721649&{}:{}&0.607310215558&{}:{}&0.002999062793&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.673663366337&{}:{}&0.324032403240&{}:{}&0.002304230423&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.444705882353&{}:{}&0.308021390374&{}:{}&0.247272727273&, \end{alignedat} \]
6 (220)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6}}&{}\approx{}&0.551020408163&{}:{}&0.489795918367&{}:{}&-0.040816326531&,\\ A^*_{\mathbf{6}}&{}\approx{}&0.000000000000&{}:{}&1.090909090909&{}:{}&-0.090909090909&,\\B^*_{\mathbf{6}}&{}\approx{}&1.080000000000&{}:{}&0.000000000000&{}:{}&-0.080000000000&,\\C^*_{\mathbf{6}}&{}\approx{}&0.529411764706&{}:{}&0.470588235294&{}:{}&0.000000000000&. \end{alignedat} \]
6 (220)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6}}&{}\approx{}&0.517808219178&{}:{}&0.448318804483&{}:{}&0.033872976339&,\\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.389690721649&{}:{}&0.607310215558&{}:{}&0.002999062793&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.673663366337&{}:{}&0.324032403240&{}:{}&0.002304230423&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.444705882353&{}:{}&0.308021390374&{}:{}&0.247272727273&,\\ A^*_{\mathbf{6}}&{}\approx{}&0.000000000000&{}:{}&1.090909090909&{}:{}&-0.090909090909&,\\B^*_{\mathbf{6}}&{}\approx{}&1.080000000000&{}:{}&0.000000000000&{}:{}&-0.080000000000&,\\C^*_{\mathbf{6}}&{}\approx{}&0.529411764706&{}:{}&0.470588235294&{}:{}&0.000000000000&. \end{alignedat} \]
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