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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1c (112)

Malfatti circles

1c (112)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.203125000000&{}:{}&1.302083333333&{}:{}&-1.505208333333&, \\ P_{\mathbf{1c}}&{}\approx{}&-0.123293357934&{}:{}&-0.158210332103&{}:{}&1.281503690037&, \\ P^-_{\mathbf{1c}}&{}\approx{}&0.166023514137&{}:{}&0.160307270629&{}:{}&0.673669215234&, \\ P^+_{\mathbf{1c}}&{}\approx{}&-0.636482855681&{}:{}&-0.723196008188&{}:{}&2.359678863869&, \\ Q_{\mathbf{1c}}&{}\approx{}&6.600000000000&{}:{}&7.000000000000&{}:{}&-12.600000000000&, \\ I^\prime_{\mathbf{1c}}&{}\approx{}&-0.700471698113&{}:{}&-0.825471698113&{}:{}&2.525943396226&, \end{alignedat} \]
\(I_{\mathbf{c}}\) Incenter
\(P_{\mathbf{1c}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{1c}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{1c}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{1c}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{1c}}\) Radical center of the Malfatti circles
1c (112)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1c}}&{}\approx{}&0.789351851852&{}:{}&-1.350308641975&{}:{}&1.560956790123&,\\B^\prime_{\mathbf{1c}}&{}\approx{}&-1.160156250000&{}:{}&0.708705357143&{}:{}&1.451450892857&,\\C^\prime_{\mathbf{1c}}&{}\approx{}&-2.062500000000&{}:{}&-2.232142857143&{}:{}&5.294642857143&, \\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.485000000000&{}:{}&-0.350000000000&{}:{}&2.835000000000&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.285576923077&{}:{}&-1.682692307692&{}:{}&2.968269230769&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.220836087244&{}:{}&-0.283377395902&{}:{}&1.504213483146&, \\ A^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.459143109541&{}:{}&0.280477031802&{}:{}&1.178666077739&,\\B^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.298773364486&{}:{}&-0.511098130841&{}:{}&1.212324766355&,\\C^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.253608417805&{}:{}&0.244876597620&{}:{}&0.501514984575&, \\ A^*_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&-1.250000000000&{}:{}&2.250000000000&,\\B^*_{\mathbf{1c}}&{}\approx{}&-1.100000000000&{}:{}&0.000000000000&{}:{}&2.100000000000&,\\C^*_{\mathbf{1c}}&{}\approx{}&0.485294117647&{}:{}&0.514705882353&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1c}}}{B^\prime_{\mathbf{1c}}}{C^\prime_{\mathbf{1c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1c}}}{B^{\prime\prime}_{\mathbf{1c}}}{C^{\prime\prime}_{\mathbf{1c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1c}}}{B^{\prime\prime\prime}_{\mathbf{1c}}}{C^{\prime\prime\prime}_{\mathbf{1c}}}\)
\(\triangle{A^*_{\mathbf{1c}}}{B^*_{\mathbf{1c}}}{C^*_{\mathbf{1c}}}\)
1c (112)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1c}}}}&{}\approx{}&-1.037037037037&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1c}}}}&{}\approx{}&-0.964285714286&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1c}}}}&{}\approx{}&-1.714285714286&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.203125000000&{}:{}&1.302083333333&{}:{}&-1.505208333333&,\\ A^\prime_{\mathbf{1c}}&{}\approx{}&0.789351851852&{}:{}&-1.350308641975&{}:{}&1.560956790123&,\\B^\prime_{\mathbf{1c}}&{}\approx{}&-1.160156250000&{}:{}&0.708705357143&{}:{}&1.451450892857&,\\C^\prime_{\mathbf{1c}}&{}\approx{}&-2.062500000000&{}:{}&-2.232142857143&{}:{}&5.294642857143&. \end{alignedat} \]
1c (112)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1c}}&{}\approx{}&-0.123293357934&{}:{}&-0.158210332103&{}:{}&1.281503690037&,\\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.485000000000&{}:{}&-0.350000000000&{}:{}&2.835000000000&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.285576923077&{}:{}&-1.682692307692&{}:{}&2.968269230769&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.220836087244&{}:{}&-0.283377395902&{}:{}&1.504213483146&. \end{alignedat} \]
1c (112)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1c}}&{}\approx{}&0.166023514137&{}:{}&0.160307270629&{}:{}&0.673669215234&,\\ A^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.459143109541&{}:{}&0.280477031802&{}:{}&1.178666077739&,\\B^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.298773364486&{}:{}&-0.511098130841&{}:{}&1.212324766355&,\\C^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.253608417805&{}:{}&0.244876597620&{}:{}&0.501514984575&. \end{alignedat} \]
1c (112)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1c}}&{}\approx{}&-0.636482855681&{}:{}&-0.723196008188&{}:{}&2.359678863869&,\\ A^\prime_{\mathbf{1c}}&{}\approx{}&0.789351851852&{}:{}&-1.350308641975&{}:{}&1.560956790123&,\\B^\prime_{\mathbf{1c}}&{}\approx{}&-1.160156250000&{}:{}&0.708705357143&{}:{}&1.451450892857&,\\C^\prime_{\mathbf{1c}}&{}\approx{}&-2.062500000000&{}:{}&-2.232142857143&{}:{}&5.294642857143&,\\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.485000000000&{}:{}&-0.350000000000&{}:{}&2.835000000000&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.285576923077&{}:{}&-1.682692307692&{}:{}&2.968269230769&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.220836087244&{}:{}&-0.283377395902&{}:{}&1.504213483146&, \end{alignedat} \]
1c (112)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1c}}&{}\approx{}&6.600000000000&{}:{}&7.000000000000&{}:{}&-12.600000000000&,\\ A^*_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&-1.250000000000&{}:{}&2.250000000000&,\\B^*_{\mathbf{1c}}&{}\approx{}&-1.100000000000&{}:{}&0.000000000000&{}:{}&2.100000000000&,\\C^*_{\mathbf{1c}}&{}\approx{}&0.485294117647&{}:{}&0.514705882353&{}:{}&0.000000000000&. \end{alignedat} \]
1c (112)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1c}}&{}\approx{}&-0.700471698113&{}:{}&-0.825471698113&{}:{}&2.525943396226&,\\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.485000000000&{}:{}&-0.350000000000&{}:{}&2.835000000000&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.285576923077&{}:{}&-1.682692307692&{}:{}&2.968269230769&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.220836087244&{}:{}&-0.283377395902&{}:{}&1.504213483146&,\\ A^*_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&-1.250000000000&{}:{}&2.250000000000&,\\B^*_{\mathbf{1c}}&{}\approx{}&-1.100000000000&{}:{}&0.000000000000&{}:{}&2.100000000000&,\\C^*_{\mathbf{1c}}&{}\approx{}&0.485294117647&{}:{}&0.514705882353&{}:{}&0.000000000000&. \end{alignedat} \]
1c (112)