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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3a (033)

Malfatti circles

3a (033)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.750000000000&{}:{}&0.811688311688&{}:{}&0.938311688312&, \\ P_{\mathbf{3a}}&{}\approx{}&-0.024137931034&{}:{}&0.338557993730&{}:{}&0.685579937304&, \\ P^-_{\mathbf{3a}}&{}\approx{}&0.098790322581&{}:{}&0.258431085044&{}:{}&0.642778592375&, \\ P^+_{\mathbf{3a}}&{}\approx{}&-0.115963855422&{}:{}&0.398411829135&{}:{}&0.717552026287&, \\ Q_{\mathbf{3a}}&{}\approx{}&0.057142857143&{}:{}&0.428571428571&{}:{}&0.514285714286&, \\ I^\prime_{\mathbf{3a}}&{}\approx{}&-0.112903225806&{}:{}&0.439882697947&{}:{}&0.673020527859&, \end{alignedat} \]
\(I_{\mathbf{a}}\) Incenter
\(P_{\mathbf{3a}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{3a}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{3a}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{3a}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{3a}}\) Radical center of the Malfatti circles
3a (033)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3a}}&{}\approx{}&0.250000000000&{}:{}&0.347866419295&{}:{}&0.402133580705&,\\B^\prime_{\mathbf{3a}}&{}\approx{}&-0.437500000000&{}:{}&0.890151515152&{}:{}&0.547348484848&,\\C^\prime_{\mathbf{3a}}&{}\approx{}&-0.194444444444&{}:{}&0.210437710438&{}:{}&0.984006734007&, \\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.269230769231&{}:{}&0.419580419580&{}:{}&0.849650349650&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.026923076923&{}:{}&0.262237762238&{}:{}&0.764685314685&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.041176470588&{}:{}&0.577540106952&{}:{}&0.463636363636&, \\ A^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.164062500000&{}:{}&0.333806818182&{}:{}&0.830255681818&,\\B^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.112385321101&{}:{}&0.156380316931&{}:{}&0.731234361968&,\\C^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.191406250000&{}:{}&0.500710227273&{}:{}&0.307883522727&, \\ A^*_{\mathbf{3a}}&{}\approx{}&0.000000000000&{}:{}&0.454545454545&{}:{}&0.545454545455&,\\B^*_{\mathbf{3a}}&{}\approx{}&0.100000000000&{}:{}&0.000000000000&{}:{}&0.900000000000&,\\C^*_{\mathbf{3a}}&{}\approx{}&0.117647058824&{}:{}&0.882352941176&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3a}}}{B^\prime_{\mathbf{3a}}}{C^\prime_{\mathbf{3a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3a}}}{B^{\prime\prime}_{\mathbf{3a}}}{C^{\prime\prime}_{\mathbf{3a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3a}}}{B^{\prime\prime\prime}_{\mathbf{3a}}}{C^{\prime\prime\prime}_{\mathbf{3a}}}\)
\(\triangle{A^*_{\mathbf{3a}}}{B^*_{\mathbf{3a}}}{C^*_{\mathbf{3a}}}\)
3a (033)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3a}}}}&{}\approx{}&0.428571428571&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3a}}}}&{}\approx{}&0.583333333333&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3a}}}}&{}\approx{}&0.259259259259&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.750000000000&{}:{}&0.811688311688&{}:{}&0.938311688312&,\\ A^\prime_{\mathbf{3a}}&{}\approx{}&0.250000000000&{}:{}&0.347866419295&{}:{}&0.402133580705&,\\B^\prime_{\mathbf{3a}}&{}\approx{}&-0.437500000000&{}:{}&0.890151515152&{}:{}&0.547348484848&,\\C^\prime_{\mathbf{3a}}&{}\approx{}&-0.194444444444&{}:{}&0.210437710438&{}:{}&0.984006734007&. \end{alignedat} \]
3a (033)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3a}}&{}\approx{}&-0.024137931034&{}:{}&0.338557993730&{}:{}&0.685579937304&,\\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.269230769231&{}:{}&0.419580419580&{}:{}&0.849650349650&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.026923076923&{}:{}&0.262237762238&{}:{}&0.764685314685&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.041176470588&{}:{}&0.577540106952&{}:{}&0.463636363636&. \end{alignedat} \]
3a (033)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3a}}&{}\approx{}&0.098790322581&{}:{}&0.258431085044&{}:{}&0.642778592375&,\\ A^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.164062500000&{}:{}&0.333806818182&{}:{}&0.830255681818&,\\B^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.112385321101&{}:{}&0.156380316931&{}:{}&0.731234361968&,\\C^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.191406250000&{}:{}&0.500710227273&{}:{}&0.307883522727&. \end{alignedat} \]
3a (033)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3a}}&{}\approx{}&-0.115963855422&{}:{}&0.398411829135&{}:{}&0.717552026287&,\\ A^\prime_{\mathbf{3a}}&{}\approx{}&0.250000000000&{}:{}&0.347866419295&{}:{}&0.402133580705&,\\B^\prime_{\mathbf{3a}}&{}\approx{}&-0.437500000000&{}:{}&0.890151515152&{}:{}&0.547348484848&,\\C^\prime_{\mathbf{3a}}&{}\approx{}&-0.194444444444&{}:{}&0.210437710438&{}:{}&0.984006734007&,\\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.269230769231&{}:{}&0.419580419580&{}:{}&0.849650349650&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.026923076923&{}:{}&0.262237762238&{}:{}&0.764685314685&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.041176470588&{}:{}&0.577540106952&{}:{}&0.463636363636&, \end{alignedat} \]
3a (033)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3a}}&{}\approx{}&0.057142857143&{}:{}&0.428571428571&{}:{}&0.514285714286&,\\ A^*_{\mathbf{3a}}&{}\approx{}&0.000000000000&{}:{}&0.454545454545&{}:{}&0.545454545455&,\\B^*_{\mathbf{3a}}&{}\approx{}&0.100000000000&{}:{}&0.000000000000&{}:{}&0.900000000000&,\\C^*_{\mathbf{3a}}&{}\approx{}&0.117647058824&{}:{}&0.882352941176&{}:{}&0.000000000000&. \end{alignedat} \]
3a (033)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3a}}&{}\approx{}&-0.112903225806&{}:{}&0.439882697947&{}:{}&0.673020527859&,\\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.269230769231&{}:{}&0.419580419580&{}:{}&0.849650349650&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.026923076923&{}:{}&0.262237762238&{}:{}&0.764685314685&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.041176470588&{}:{}&0.577540106952&{}:{}&0.463636363636&,\\ A^*_{\mathbf{3a}}&{}\approx{}&0.000000000000&{}:{}&0.454545454545&{}:{}&0.545454545455&,\\B^*_{\mathbf{3a}}&{}\approx{}&0.100000000000&{}:{}&0.000000000000&{}:{}&0.900000000000&,\\C^*_{\mathbf{3a}}&{}\approx{}&0.117647058824&{}:{}&0.882352941176&{}:{}&0.000000000000&. \end{alignedat} \]
3a (033)