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 > 6b (321)

6b (321)

Malfatti circles

6b (321)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.855555555556&{}:{}&-0.925925925926&{}:{}&1.070370370370&, \\ P_{\mathbf{6b}}&{}\approx{}&-0.040286123033&{}:{}&1.046828803052&{}:{}&-0.006542680019&, \\ P^-_{\mathbf{6b}}&{}\approx{}&0.019962636776&{}:{}&0.914153545058&{}:{}&0.065883818166&, \\ P^+_{\mathbf{6b}}&{}\approx{}&-0.109898242368&{}:{}&1.200123342584&{}:{}&-0.090225100216&, \\ Q_{\mathbf{6b}}&{}\approx{}&-0.536585365854&{}:{}&1.365853658537&{}:{}&0.170731707317&, \\ I^\prime_{\mathbf{6b}}&{}\approx{}&-0.259587020649&{}:{}&1.376597836775&{}:{}&-0.117010816126&, \end{alignedat} \]
\(I_{\mathbf{b}}\) Incenter
\(P_{\mathbf{6b}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{6b}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{6b}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{6b}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{6b}}\) Radical center of the Malfatti circles
6b (321)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6b}}&{}\approx{}&1.947916666667&{}:{}&6.076388888889&{}:{}&-7.024305555556&,\\B^\prime_{\mathbf{6b}}&{}\approx{}&-0.073333333333&{}:{}&1.165079365079&{}:{}&-0.091746031746&,\\C^\prime_{\mathbf{6b}}&{}\approx{}&-0.814814814815&{}:{}&0.881834215168&{}:{}&0.932980599647&, \\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.134556574924&{}:{}&1.141692150866&{}:{}&-0.007135575943&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.464686468647&{}:{}&1.540154015402&{}:{}&-0.075467546755&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.047273704002&{}:{}&1.228400035813&{}:{}&-0.181126331811&, \\ A^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.062373430539&{}:{}&0.990954502498&{}:{}&0.071418928041&,\\B^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.134774774775&{}:{}&0.420420420420&{}:{}&0.444804804805&,\\C^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.023155027241&{}:{}&1.060343404326&{}:{}&-0.083498431567&, \\ A^*_{\mathbf{6b}}&{}\approx{}&0.000000000000&{}:{}&0.888888888889&{}:{}&0.111111111111&,\\B^*_{\mathbf{6b}}&{}\approx{}&1.466666666667&{}:{}&0.000000000000&{}:{}&-0.466666666667&,\\C^*_{\mathbf{6b}}&{}\approx{}&-0.647058823529&{}:{}&1.647058823529&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6b}}}{B^\prime_{\mathbf{6b}}}{C^\prime_{\mathbf{6b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6b}}}{B^{\prime\prime}_{\mathbf{6b}}}{C^{\prime\prime}_{\mathbf{6b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6b}}}{B^{\prime\prime\prime}_{\mathbf{6b}}}{C^{\prime\prime\prime}_{\mathbf{6b}}}\)
\(\triangle{A^*_{\mathbf{6b}}}{B^*_{\mathbf{6b}}}{C^*_{\mathbf{6b}}}\)
6b (321)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6b}}}}&{}\approx{}&-6.562500000000&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6b}}}}&{}\approx{}&-0.085714285714&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6b}}}}&{}\approx{}&-0.952380952381&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.855555555556&{}:{}&-0.925925925926&{}:{}&1.070370370370&,\\ A^\prime_{\mathbf{6b}}&{}\approx{}&1.947916666667&{}:{}&6.076388888889&{}:{}&-7.024305555556&,\\B^\prime_{\mathbf{6b}}&{}\approx{}&-0.073333333333&{}:{}&1.165079365079&{}:{}&-0.091746031746&,\\C^\prime_{\mathbf{6b}}&{}\approx{}&-0.814814814815&{}:{}&0.881834215168&{}:{}&0.932980599647&. \end{alignedat} \]
6b (321)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6b}}&{}\approx{}&-0.040286123033&{}:{}&1.046828803052&{}:{}&-0.006542680019&,\\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.134556574924&{}:{}&1.141692150866&{}:{}&-0.007135575943&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.464686468647&{}:{}&1.540154015402&{}:{}&-0.075467546755&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.047273704002&{}:{}&1.228400035813&{}:{}&-0.181126331811&. \end{alignedat} \]
6b (321)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6b}}&{}\approx{}&0.019962636776&{}:{}&0.914153545058&{}:{}&0.065883818166&,\\ A^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.062373430539&{}:{}&0.990954502498&{}:{}&0.071418928041&,\\B^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.134774774775&{}:{}&0.420420420420&{}:{}&0.444804804805&,\\C^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.023155027241&{}:{}&1.060343404326&{}:{}&-0.083498431567&. \end{alignedat} \]
6b (321)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6b}}&{}\approx{}&-0.109898242368&{}:{}&1.200123342584&{}:{}&-0.090225100216&,\\ A^\prime_{\mathbf{6b}}&{}\approx{}&1.947916666667&{}:{}&6.076388888889&{}:{}&-7.024305555556&,\\B^\prime_{\mathbf{6b}}&{}\approx{}&-0.073333333333&{}:{}&1.165079365079&{}:{}&-0.091746031746&,\\C^\prime_{\mathbf{6b}}&{}\approx{}&-0.814814814815&{}:{}&0.881834215168&{}:{}&0.932980599647&,\\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.134556574924&{}:{}&1.141692150866&{}:{}&-0.007135575943&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.464686468647&{}:{}&1.540154015402&{}:{}&-0.075467546755&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.047273704002&{}:{}&1.228400035813&{}:{}&-0.181126331811&, \end{alignedat} \]
6b (321)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6b}}&{}\approx{}&-0.536585365854&{}:{}&1.365853658537&{}:{}&0.170731707317&,\\ A^*_{\mathbf{6b}}&{}\approx{}&0.000000000000&{}:{}&0.888888888889&{}:{}&0.111111111111&,\\B^*_{\mathbf{6b}}&{}\approx{}&1.466666666667&{}:{}&0.000000000000&{}:{}&-0.466666666667&,\\C^*_{\mathbf{6b}}&{}\approx{}&-0.647058823529&{}:{}&1.647058823529&{}:{}&0.000000000000&. \end{alignedat} \]
6b (321)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6b}}&{}\approx{}&-0.259587020649&{}:{}&1.376597836775&{}:{}&-0.117010816126&,\\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.134556574924&{}:{}&1.141692150866&{}:{}&-0.007135575943&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.464686468647&{}:{}&1.540154015402&{}:{}&-0.075467546755&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.047273704002&{}:{}&1.228400035813&{}:{}&-0.181126331811&,\\ A^*_{\mathbf{6b}}&{}\approx{}&0.000000000000&{}:{}&0.888888888889&{}:{}&0.111111111111&,\\B^*_{\mathbf{6b}}&{}\approx{}&1.466666666667&{}:{}&0.000000000000&{}:{}&-0.466666666667&,\\C^*_{\mathbf{6b}}&{}\approx{}&-0.647058823529&{}:{}&1.647058823529&{}:{}&0.000000000000&. \end{alignedat} \]
6b (321)