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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6c (330)

Malfatti circles

6c (330)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.203125000000&{}:{}&1.302083333333&{}:{}&-1.505208333333&, \\ P_{\mathbf{6c}}&{}\approx{}&0.464062500000&{}:{}&0.595486111111&{}:{}&-0.059548611111&, \\ P^-_{\mathbf{6c}}&{}\approx{}&0.366450471698&{}:{}&0.502161949686&{}:{}&0.131387578616&, \\ P^+_{\mathbf{6c}}&{}\approx{}&0.541277985075&{}:{}&0.669309701493&{}:{}&-0.210587686567&, \\ Q_{\mathbf{6c}}&{}\approx{}&0.440000000000&{}:{}&0.466666666667&{}:{}&0.093333333333&, \\ I^\prime_{\mathbf{6c}}&{}\approx{}&0.562500000000&{}:{}&0.662878787879&{}:{}&-0.225378787879&, \end{alignedat} \]
\(I_{\mathbf{c}}\) Incenter
\(P_{\mathbf{6c}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{6c}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{6c}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{6c}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{6c}}\) Radical center of the Malfatti circles
6c (330)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6c}}&{}\approx{}&1.078993055556&{}:{}&0.506365740741&{}:{}&-0.585358796296&,\\B^\prime_{\mathbf{6c}}&{}\approx{}&0.343750000000&{}:{}&1.086309523810&{}:{}&-0.430059523810&,\\C^\prime_{\mathbf{6c}}&{}\approx{}&0.343750000000&{}:{}&0.372023809524&{}:{}&0.284226190476&, \\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.343750000000&{}:{}&0.729166666667&{}:{}&-0.072916666667&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.655147058824&{}:{}&0.428921568627&{}:{}&-0.084068627451&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.655147058824&{}:{}&0.840686274510&{}:{}&-0.495833333333&, \\ A^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.200520833333&{}:{}&0.633680555556&{}:{}&0.165798611111&,\\B^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.547095070423&{}:{}&0.256748826291&{}:{}&0.196156103286&,\\C^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.547095070423&{}:{}&0.749706572770&{}:{}&-0.296801643192&, \\ A^*_{\mathbf{6c}}&{}\approx{}&0.000000000000&{}:{}&0.833333333333&{}:{}&0.166666666667&,\\B^*_{\mathbf{6c}}&{}\approx{}&0.825000000000&{}:{}&0.000000000000&{}:{}&0.175000000000&,\\C^*_{\mathbf{6c}}&{}\approx{}&0.485294117647&{}:{}&0.514705882353&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6c}}}{B^\prime_{\mathbf{6c}}}{C^\prime_{\mathbf{6c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6c}}}{B^{\prime\prime}_{\mathbf{6c}}}{C^{\prime\prime}_{\mathbf{6c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6c}}}{B^{\prime\prime\prime}_{\mathbf{6c}}}{C^{\prime\prime\prime}_{\mathbf{6c}}}\)
\(\triangle{A^*_{\mathbf{6c}}}{B^*_{\mathbf{6c}}}{C^*_{\mathbf{6c}}}\)
6c (330)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6c}}}}&{}\approx{}&0.388888888889&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6c}}}}&{}\approx{}&0.285714285714&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6c}}}}&{}\approx{}&0.285714285714&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.203125000000&{}:{}&1.302083333333&{}:{}&-1.505208333333&,\\ A^\prime_{\mathbf{6c}}&{}\approx{}&1.078993055556&{}:{}&0.506365740741&{}:{}&-0.585358796296&,\\B^\prime_{\mathbf{6c}}&{}\approx{}&0.343750000000&{}:{}&1.086309523810&{}:{}&-0.430059523810&,\\C^\prime_{\mathbf{6c}}&{}\approx{}&0.343750000000&{}:{}&0.372023809524&{}:{}&0.284226190476&. \end{alignedat} \]
6c (330)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6c}}&{}\approx{}&0.464062500000&{}:{}&0.595486111111&{}:{}&-0.059548611111&,\\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.343750000000&{}:{}&0.729166666667&{}:{}&-0.072916666667&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.655147058824&{}:{}&0.428921568627&{}:{}&-0.084068627451&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.655147058824&{}:{}&0.840686274510&{}:{}&-0.495833333333&. \end{alignedat} \]
6c (330)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6c}}&{}\approx{}&0.366450471698&{}:{}&0.502161949686&{}:{}&0.131387578616&,\\ A^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.200520833333&{}:{}&0.633680555556&{}:{}&0.165798611111&,\\B^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.547095070423&{}:{}&0.256748826291&{}:{}&0.196156103286&,\\C^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.547095070423&{}:{}&0.749706572770&{}:{}&-0.296801643192&. \end{alignedat} \]
6c (330)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6c}}&{}\approx{}&0.541277985075&{}:{}&0.669309701493&{}:{}&-0.210587686567&,\\ A^\prime_{\mathbf{6c}}&{}\approx{}&1.078993055556&{}:{}&0.506365740741&{}:{}&-0.585358796296&,\\B^\prime_{\mathbf{6c}}&{}\approx{}&0.343750000000&{}:{}&1.086309523810&{}:{}&-0.430059523810&,\\C^\prime_{\mathbf{6c}}&{}\approx{}&0.343750000000&{}:{}&0.372023809524&{}:{}&0.284226190476&,\\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.343750000000&{}:{}&0.729166666667&{}:{}&-0.072916666667&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.655147058824&{}:{}&0.428921568627&{}:{}&-0.084068627451&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.655147058824&{}:{}&0.840686274510&{}:{}&-0.495833333333&, \end{alignedat} \]
6c (330)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6c}}&{}\approx{}&0.440000000000&{}:{}&0.466666666667&{}:{}&0.093333333333&,\\ A^*_{\mathbf{6c}}&{}\approx{}&0.000000000000&{}:{}&0.833333333333&{}:{}&0.166666666667&,\\B^*_{\mathbf{6c}}&{}\approx{}&0.825000000000&{}:{}&0.000000000000&{}:{}&0.175000000000&,\\C^*_{\mathbf{6c}}&{}\approx{}&0.485294117647&{}:{}&0.514705882353&{}:{}&0.000000000000&. \end{alignedat} \]
6c (330)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6c}}&{}\approx{}&0.562500000000&{}:{}&0.662878787879&{}:{}&-0.225378787879&,\\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.343750000000&{}:{}&0.729166666667&{}:{}&-0.072916666667&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.655147058824&{}:{}&0.428921568627&{}:{}&-0.084068627451&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.655147058824&{}:{}&0.840686274510&{}:{}&-0.495833333333&,\\ A^*_{\mathbf{6c}}&{}\approx{}&0.000000000000&{}:{}&0.833333333333&{}:{}&0.166666666667&,\\B^*_{\mathbf{6c}}&{}\approx{}&0.825000000000&{}:{}&0.000000000000&{}:{}&0.175000000000&,\\C^*_{\mathbf{6c}}&{}\approx{}&0.485294117647&{}:{}&0.514705882353&{}:{}&0.000000000000&. \end{alignedat} \]
6c (330)