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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2c (130)

Malfatti circles

2c (130)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.203125000000&{}:{}&1.302083333333&{}:{}&-1.505208333333&, \\ P_{\mathbf{2c}}&{}\approx{}&0.963407109245&{}:{}&0.160217521683&{}:{}&-0.123624630928&, \\ P^-_{\mathbf{2c}}&{}\approx{}&0.940508756191&{}:{}&0.051144117052&{}:{}&0.008347126756&, \\ P^+_{\mathbf{2c}}&{}\approx{}&0.982632553255&{}:{}&0.251795489264&{}:{}&-0.234428042519&, \\ Q_{\mathbf{2c}}&{}\approx{}&1.204379562044&{}:{}&-0.459854014599&{}:{}&0.255474452555&, \\ I^\prime_{\mathbf{2c}}&{}\approx{}&0.976973684211&{}:{}&0.414473684211&{}:{}&-0.391447368421&, \end{alignedat} \]
\(I_{\mathbf{c}}\) Incenter
\(P_{\mathbf{2c}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{2c}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{2c}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{2c}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{2c}}\) Radical center of the Malfatti circles
2c (130)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{2c}}&{}\approx{}&1.021064814815&{}:{}&0.135030864198&{}:{}&-0.156095679012&,\\B^\prime_{\mathbf{2c}}&{}\approx{}&5.156250000000&{}:{}&2.294642857143&{}:{}&-6.450892857143&,\\C^\prime_{\mathbf{2c}}&{}\approx{}&0.464062500000&{}:{}&0.502232142857&{}:{}&0.033705357143&, \\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.851490825688&{}:{}&0.650229357798&{}:{}&-0.501720183486&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.903040540541&{}:{}&0.212837837838&{}:{}&-0.115878378378&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.118762054002&{}:{}&0.186053519769&{}:{}&-0.304815573770&, \\ A^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.658912271805&{}:{}&0.293230223124&{}:{}&0.047857505071&,\\B^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.876331822463&{}:{}&0.115890628679&{}:{}&0.007777548858&,\\C^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.109257644850&{}:{}&0.060320547210&{}:{}&-0.169578192060&, \\ A^*_{\mathbf{2c}}&{}\approx{}&0.000000000000&{}:{}&2.250000000000&{}:{}&-1.250000000000&,\\B^*_{\mathbf{2c}}&{}\approx{}&0.825000000000&{}:{}&0.000000000000&{}:{}&0.175000000000&,\\C^*_{\mathbf{2c}}&{}\approx{}&1.617647058824&{}:{}&-0.617647058824&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{2c}}}{B^\prime_{\mathbf{2c}}}{C^\prime_{\mathbf{2c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{2c}}}{B^{\prime\prime}_{\mathbf{2c}}}{C^{\prime\prime}_{\mathbf{2c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{2c}}}{B^{\prime\prime\prime}_{\mathbf{2c}}}{C^{\prime\prime\prime}_{\mathbf{2c}}}\)
\(\triangle{A^*_{\mathbf{2c}}}{B^*_{\mathbf{2c}}}{C^*_{\mathbf{2c}}}\)
2c (130)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{2c}}}}&{}\approx{}&0.103703703704&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{2c}}}}&{}\approx{}&4.285714285714&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{2c}}}}&{}\approx{}&0.385714285714&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.203125000000&{}:{}&1.302083333333&{}:{}&-1.505208333333&,\\ A^\prime_{\mathbf{2c}}&{}\approx{}&1.021064814815&{}:{}&0.135030864198&{}:{}&-0.156095679012&,\\B^\prime_{\mathbf{2c}}&{}\approx{}&5.156250000000&{}:{}&2.294642857143&{}:{}&-6.450892857143&,\\C^\prime_{\mathbf{2c}}&{}\approx{}&0.464062500000&{}:{}&0.502232142857&{}:{}&0.033705357143&. \end{alignedat} \]
2c (130)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{2c}}&{}\approx{}&0.963407109245&{}:{}&0.160217521683&{}:{}&-0.123624630928&,\\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.851490825688&{}:{}&0.650229357798&{}:{}&-0.501720183486&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.903040540541&{}:{}&0.212837837838&{}:{}&-0.115878378378&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.118762054002&{}:{}&0.186053519769&{}:{}&-0.304815573770&. \end{alignedat} \]
2c (130)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{2c}}&{}\approx{}&0.940508756191&{}:{}&0.051144117052&{}:{}&0.008347126756&,\\ A^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.658912271805&{}:{}&0.293230223124&{}:{}&0.047857505071&,\\B^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.876331822463&{}:{}&0.115890628679&{}:{}&0.007777548858&,\\C^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.109257644850&{}:{}&0.060320547210&{}:{}&-0.169578192060&. \end{alignedat} \]
2c (130)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{2c}}&{}\approx{}&0.982632553255&{}:{}&0.251795489264&{}:{}&-0.234428042519&,\\ A^\prime_{\mathbf{2c}}&{}\approx{}&1.021064814815&{}:{}&0.135030864198&{}:{}&-0.156095679012&,\\B^\prime_{\mathbf{2c}}&{}\approx{}&5.156250000000&{}:{}&2.294642857143&{}:{}&-6.450892857143&,\\C^\prime_{\mathbf{2c}}&{}\approx{}&0.464062500000&{}:{}&0.502232142857&{}:{}&0.033705357143&,\\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.851490825688&{}:{}&0.650229357798&{}:{}&-0.501720183486&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.903040540541&{}:{}&0.212837837838&{}:{}&-0.115878378378&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.118762054002&{}:{}&0.186053519769&{}:{}&-0.304815573770&, \end{alignedat} \]
2c (130)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{2c}}&{}\approx{}&1.204379562044&{}:{}&-0.459854014599&{}:{}&0.255474452555&,\\ A^*_{\mathbf{2c}}&{}\approx{}&0.000000000000&{}:{}&2.250000000000&{}:{}&-1.250000000000&,\\B^*_{\mathbf{2c}}&{}\approx{}&0.825000000000&{}:{}&0.000000000000&{}:{}&0.175000000000&,\\C^*_{\mathbf{2c}}&{}\approx{}&1.617647058824&{}:{}&-0.617647058824&{}:{}&0.000000000000&. \end{alignedat} \]
2c (130)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{2c}}&{}\approx{}&0.976973684211&{}:{}&0.414473684211&{}:{}&-0.391447368421&,\\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.851490825688&{}:{}&0.650229357798&{}:{}&-0.501720183486&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.903040540541&{}:{}&0.212837837838&{}:{}&-0.115878378378&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.118762054002&{}:{}&0.186053519769&{}:{}&-0.304815573770&,\\ A^*_{\mathbf{2c}}&{}\approx{}&0.000000000000&{}:{}&2.250000000000&{}:{}&-1.250000000000&,\\B^*_{\mathbf{2c}}&{}\approx{}&0.825000000000&{}:{}&0.000000000000&{}:{}&0.175000000000&,\\C^*_{\mathbf{2c}}&{}\approx{}&1.617647058824&{}:{}&-0.617647058824&{}:{}&0.000000000000&. \end{alignedat} \]
2c (130)