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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5c (312)

Malfatti circles

5c (312)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.203125000000&{}:{}&1.302083333333&{}:{}&-1.505208333333&, \\ P_{\mathbf{5c}}&{}\approx{}&-0.108370113493&{}:{}&-0.018022278268&{}:{}&1.126392391761&, \\ P^-_{\mathbf{5c}}&{}\approx{}&-0.001991599073&{}:{}&0.089054654307&{}:{}&0.912936944766&, \\ P^+_{\mathbf{5c}}&{}\approx{}&-0.235347510373&{}:{}&-0.145833333333&{}:{}&1.381180843707&, \\ Q_{\mathbf{5c}}&{}\approx{}&-0.774647887324&{}:{}&0.295774647887&{}:{}&1.478873239437&, \\ I^\prime_{\mathbf{5c}}&{}\approx{}&-0.458333333333&{}:{}&-0.194444444444&{}:{}&1.652777777778&, \end{alignedat} \]
\(I_{\mathbf{c}}\) Incenter
\(P_{\mathbf{5c}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{5c}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{5c}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{5c}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{5c}}\) Radical center of the Malfatti circles
5c (312)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5c}}&{}\approx{}&-0.137500000000&{}:{}&-7.291666666667&{}:{}&8.429166666667&,\\B^\prime_{\mathbf{5c}}&{}\approx{}&-0.859375000000&{}:{}&0.784226190476&{}:{}&1.075148809524&,\\C^\prime_{\mathbf{5c}}&{}\approx{}&-0.137500000000&{}:{}&-0.148809523810&{}:{}&1.286309523810&, \\ A^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.237068965517&{}:{}&-0.020114942529&{}:{}&1.257183908046&,\\B^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.137500000000&{}:{}&-0.291666666667&{}:{}&1.429166666667&,\\C^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.777714932127&{}:{}&-0.129336349925&{}:{}&1.907051282051&, \\ A^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.107903587444&{}:{}&0.098467862481&{}:{}&1.009435724963&,\\B^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.002473021583&{}:{}&-0.131145083933&{}:{}&1.133618105516&,\\C^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.009522160665&{}:{}&0.425784856879&{}:{}&0.583737303786&, \\ A^*_{\mathbf{5c}}&{}\approx{}&0.000000000000&{}:{}&0.166666666667&{}:{}&0.833333333333&,\\B^*_{\mathbf{5c}}&{}\approx{}&-1.100000000000&{}:{}&0.000000000000&{}:{}&2.100000000000&,\\C^*_{\mathbf{5c}}&{}\approx{}&1.617647058824&{}:{}&-0.617647058824&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5c}}}{B^\prime_{\mathbf{5c}}}{C^\prime_{\mathbf{5c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5c}}}{B^{\prime\prime}_{\mathbf{5c}}}{C^{\prime\prime}_{\mathbf{5c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5c}}}{B^{\prime\prime\prime}_{\mathbf{5c}}}{C^{\prime\prime\prime}_{\mathbf{5c}}}\)
\(\triangle{A^*_{\mathbf{5c}}}{B^*_{\mathbf{5c}}}{C^*_{\mathbf{5c}}}\)
5c (312)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5c}}}}&{}\approx{}&-5.600000000000&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5c}}}}&{}\approx{}&-0.714285714286&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5c}}}}&{}\approx{}&-0.114285714286&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.203125000000&{}:{}&1.302083333333&{}:{}&-1.505208333333&,\\ A^\prime_{\mathbf{5c}}&{}\approx{}&-0.137500000000&{}:{}&-7.291666666667&{}:{}&8.429166666667&,\\B^\prime_{\mathbf{5c}}&{}\approx{}&-0.859375000000&{}:{}&0.784226190476&{}:{}&1.075148809524&,\\C^\prime_{\mathbf{5c}}&{}\approx{}&-0.137500000000&{}:{}&-0.148809523810&{}:{}&1.286309523810&. \end{alignedat} \]
5c (312)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5c}}&{}\approx{}&-0.108370113493&{}:{}&-0.018022278268&{}:{}&1.126392391761&,\\ A^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.237068965517&{}:{}&-0.020114942529&{}:{}&1.257183908046&,\\B^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.137500000000&{}:{}&-0.291666666667&{}:{}&1.429166666667&,\\C^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.777714932127&{}:{}&-0.129336349925&{}:{}&1.907051282051&. \end{alignedat} \]
5c (312)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5c}}&{}\approx{}&-0.001991599073&{}:{}&0.089054654307&{}:{}&0.912936944766&,\\ A^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.107903587444&{}:{}&0.098467862481&{}:{}&1.009435724963&,\\B^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.002473021583&{}:{}&-0.131145083933&{}:{}&1.133618105516&,\\C^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.009522160665&{}:{}&0.425784856879&{}:{}&0.583737303786&. \end{alignedat} \]
5c (312)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5c}}&{}\approx{}&-0.235347510373&{}:{}&-0.145833333333&{}:{}&1.381180843707&,\\ A^\prime_{\mathbf{5c}}&{}\approx{}&-0.137500000000&{}:{}&-7.291666666667&{}:{}&8.429166666667&,\\B^\prime_{\mathbf{5c}}&{}\approx{}&-0.859375000000&{}:{}&0.784226190476&{}:{}&1.075148809524&,\\C^\prime_{\mathbf{5c}}&{}\approx{}&-0.137500000000&{}:{}&-0.148809523810&{}:{}&1.286309523810&,\\ A^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.237068965517&{}:{}&-0.020114942529&{}:{}&1.257183908046&,\\B^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.137500000000&{}:{}&-0.291666666667&{}:{}&1.429166666667&,\\C^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.777714932127&{}:{}&-0.129336349925&{}:{}&1.907051282051&, \end{alignedat} \]
5c (312)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5c}}&{}\approx{}&-0.774647887324&{}:{}&0.295774647887&{}:{}&1.478873239437&,\\ A^*_{\mathbf{5c}}&{}\approx{}&0.000000000000&{}:{}&0.166666666667&{}:{}&0.833333333333&,\\B^*_{\mathbf{5c}}&{}\approx{}&-1.100000000000&{}:{}&0.000000000000&{}:{}&2.100000000000&,\\C^*_{\mathbf{5c}}&{}\approx{}&1.617647058824&{}:{}&-0.617647058824&{}:{}&0.000000000000&. \end{alignedat} \]
5c (312)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5c}}&{}\approx{}&-0.458333333333&{}:{}&-0.194444444444&{}:{}&1.652777777778&,\\ A^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.237068965517&{}:{}&-0.020114942529&{}:{}&1.257183908046&,\\B^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.137500000000&{}:{}&-0.291666666667&{}:{}&1.429166666667&,\\C^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.777714932127&{}:{}&-0.129336349925&{}:{}&1.907051282051&,\\ A^*_{\mathbf{5c}}&{}\approx{}&0.000000000000&{}:{}&0.166666666667&{}:{}&0.833333333333&,\\B^*_{\mathbf{5c}}&{}\approx{}&-1.100000000000&{}:{}&0.000000000000&{}:{}&2.100000000000&,\\C^*_{\mathbf{5c}}&{}\approx{}&1.617647058824&{}:{}&-0.617647058824&{}:{}&0.000000000000&. \end{alignedat} \]
5c (312)