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 > 0b (101)

0b (101)

Malfatti circles

0b (101)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.855555555556&{}:{}&-0.925925925926&{}:{}&1.070370370370&, \\ P_{\mathbf{b}}&{}\approx{}&0.729196337742&{}:{}&-0.451339437097&{}:{}&0.722143099356&, \\ P^-_{\mathbf{b}}&{}\approx{}&0.697324840764&{}:{}&-0.331634819533&{}:{}&0.634309978769&, \\ P^+_{\mathbf{b}}&{}\approx{}&0.750381033023&{}:{}&-0.530906011854&{}:{}&0.780524978831&, \\ Q_{\mathbf{b}}&{}\approx{}&0.717948717949&{}:{}&-0.282051282051&{}:{}&0.564102564103&, \\ I^\prime_{\mathbf{b}}&{}\approx{}&0.772413793103&{}:{}&-0.632183908046&{}:{}&0.859770114943&, \end{alignedat} \]
\(I_{\mathbf{b}}\) Incenter
\(P_{\mathbf{b}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{b}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{b}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{b}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{b}}\) Radical center of the Malfatti circles
0b (101)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{b}}&{}\approx{}&0.950347222222&{}:{}&-0.318287037037&{}:{}&0.367939814815&,\\B^\prime_{\mathbf{b}}&{}\approx{}&1.400000000000&{}:{}&-2.151515151515&{}:{}&1.751515151515&,\\C^\prime_{\mathbf{b}}&{}\approx{}&0.350000000000&{}:{}&-0.378787878788&{}:{}&1.028787878788&, \\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.560000000000&{}:{}&-0.733333333333&{}:{}&1.173333333333&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.676226415094&{}:{}&-0.345911949686&{}:{}&0.669685534591&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&1.028407460545&{}:{}&-0.636537541846&{}:{}&0.608130081301&, \\ A^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.416216216216&{}:{}&-0.639639639640&{}:{}&1.223423423423&,\\B^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.635034802784&{}:{}&-0.212683681361&{}:{}&0.577648878577&,\\C^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&1.096993987976&{}:{}&-0.521710086840&{}:{}&0.424716098864&, \\ A^*_{\mathbf{b}}&{}\approx{}&0.000000000000&{}:{}&-1.000000000000&{}:{}&2.000000000000&,\\B^*_{\mathbf{b}}&{}\approx{}&0.560000000000&{}:{}&0.000000000000&{}:{}&0.440000000000&,\\C^*_{\mathbf{b}}&{}\approx{}&1.647058823529&{}:{}&-0.647058823529&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{b}}}{B^\prime_{\mathbf{b}}}{C^\prime_{\mathbf{b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{b}}}{B^{\prime\prime}_{\mathbf{b}}}{C^{\prime\prime}_{\mathbf{b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{b}}}{B^{\prime\prime\prime}_{\mathbf{b}}}{C^{\prime\prime\prime}_{\mathbf{b}}}\)
\(\triangle{A^*_{\mathbf{b}}}{B^*_{\mathbf{b}}}{C^*_{\mathbf{b}}}\)
0b (101)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{b}}}}&{}\approx{}&0.343750000000&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{b}}}}&{}\approx{}&1.636363636364&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{b}}}}&{}\approx{}&0.409090909091&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.855555555556&{}:{}&-0.925925925926&{}:{}&1.070370370370&,\\ A^\prime_{\mathbf{b}}&{}\approx{}&0.950347222222&{}:{}&-0.318287037037&{}:{}&0.367939814815&,\\B^\prime_{\mathbf{b}}&{}\approx{}&1.400000000000&{}:{}&-2.151515151515&{}:{}&1.751515151515&,\\C^\prime_{\mathbf{b}}&{}\approx{}&0.350000000000&{}:{}&-0.378787878788&{}:{}&1.028787878788&. \end{alignedat} \]
0b (101)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{b}}&{}\approx{}&0.729196337742&{}:{}&-0.451339437097&{}:{}&0.722143099356&,\\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.560000000000&{}:{}&-0.733333333333&{}:{}&1.173333333333&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.676226415094&{}:{}&-0.345911949686&{}:{}&0.669685534591&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&1.028407460545&{}:{}&-0.636537541846&{}:{}&0.608130081301&. \end{alignedat} \]
0b (101)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{b}}&{}\approx{}&0.697324840764&{}:{}&-0.331634819533&{}:{}&0.634309978769&,\\ A^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.416216216216&{}:{}&-0.639639639640&{}:{}&1.223423423423&,\\B^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.635034802784&{}:{}&-0.212683681361&{}:{}&0.577648878577&,\\C^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&1.096993987976&{}:{}&-0.521710086840&{}:{}&0.424716098864&. \end{alignedat} \]
0b (101)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{b}}&{}\approx{}&0.750381033023&{}:{}&-0.530906011854&{}:{}&0.780524978831&,\\ A^\prime_{\mathbf{b}}&{}\approx{}&0.950347222222&{}:{}&-0.318287037037&{}:{}&0.367939814815&,\\B^\prime_{\mathbf{b}}&{}\approx{}&1.400000000000&{}:{}&-2.151515151515&{}:{}&1.751515151515&,\\C^\prime_{\mathbf{b}}&{}\approx{}&0.350000000000&{}:{}&-0.378787878788&{}:{}&1.028787878788&,\\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.560000000000&{}:{}&-0.733333333333&{}:{}&1.173333333333&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.676226415094&{}:{}&-0.345911949686&{}:{}&0.669685534591&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&1.028407460545&{}:{}&-0.636537541846&{}:{}&0.608130081301&, \end{alignedat} \]
0b (101)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{b}}&{}\approx{}&0.717948717949&{}:{}&-0.282051282051&{}:{}&0.564102564103&,\\ A^*_{\mathbf{b}}&{}\approx{}&0.000000000000&{}:{}&-1.000000000000&{}:{}&2.000000000000&,\\B^*_{\mathbf{b}}&{}\approx{}&0.560000000000&{}:{}&0.000000000000&{}:{}&0.440000000000&,\\C^*_{\mathbf{b}}&{}\approx{}&1.647058823529&{}:{}&-0.647058823529&{}:{}&0.000000000000&. \end{alignedat} \]
0b (101)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{b}}&{}\approx{}&0.772413793103&{}:{}&-0.632183908046&{}:{}&0.859770114943&,\\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.560000000000&{}:{}&-0.733333333333&{}:{}&1.173333333333&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.676226415094&{}:{}&-0.345911949686&{}:{}&0.669685534591&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&1.028407460545&{}:{}&-0.636537541846&{}:{}&0.608130081301&,\\ A^*_{\mathbf{b}}&{}\approx{}&0.000000000000&{}:{}&-1.000000000000&{}:{}&2.000000000000&,\\B^*_{\mathbf{b}}&{}\approx{}&0.560000000000&{}:{}&0.000000000000&{}:{}&0.440000000000&,\\C^*_{\mathbf{b}}&{}\approx{}&1.647058823529&{}:{}&-0.647058823529&{}:{}&0.000000000000&. \end{alignedat} \]
0b (101)