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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2a (031)

Malfatti circles

2a (031)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.750000000000&{}:{}&0.811688311688&{}:{}&0.938311688312&, \\ P_{\mathbf{2a}}&{}\approx{}&-0.034222708945&{}:{}&0.062208884749&{}:{}&0.972013824197&, \\ P^-_{\mathbf{2a}}&{}\approx{}&0.022911739411&{}:{}&0.002384279878&{}:{}&0.974703980711&, \\ P^+_{\mathbf{2a}}&{}\approx{}&-0.083491702976&{}:{}&0.113797692990&{}:{}&0.969694009986&, \\ Q_{\mathbf{2a}}&{}\approx{}&0.136986301370&{}:{}&-0.369863013699&{}:{}&1.232876712329&, \\ I^\prime_{\mathbf{2a}}&{}\approx{}&-0.157142857143&{}:{}&0.220408163265&{}:{}&0.936734693878&, \end{alignedat} \]
\(I_{\mathbf{a}}\) Incenter
\(P_{\mathbf{2a}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{2a}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{2a}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{2a}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{2a}}\) Radical center of the Malfatti circles
2a (031)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{2a}}&{}\approx{}&0.140909090909&{}:{}&0.398465171192&{}:{}&0.460625737898&,\\B^\prime_{\mathbf{2a}}&{}\approx{}&-3.437500000000&{}:{}&0.136904761905&{}:{}&4.300595238095&,\\C^\prime_{\mathbf{2a}}&{}\approx{}&-0.061111111111&{}:{}&0.066137566138&{}:{}&0.994973544974&, \\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.120087336245&{}:{}&0.067373674361&{}:{}&1.052713661884&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.032352941176&{}:{}&0.113445378151&{}:{}&0.918907563025&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.205285159749&{}:{}&0.373160431685&{}:{}&0.832124728064&, \\ A^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.065268987342&{}:{}&0.002599457505&{}:{}&1.062669529837&,\\B^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.021565899317&{}:{}&0.060984424126&{}:{}&0.917449676557&,\\C^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.228674892704&{}:{}&0.023796750460&{}:{}&0.747528356836&, \\ A^*_{\mathbf{2a}}&{}\approx{}&0.000000000000&{}:{}&-0.428571428571&{}:{}&1.428571428571&,\\B^*_{\mathbf{2a}}&{}\approx{}&0.100000000000&{}:{}&0.000000000000&{}:{}&0.900000000000&,\\C^*_{\mathbf{2a}}&{}\approx{}&-0.588235294118&{}:{}&1.588235294118&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{2a}}}{B^\prime_{\mathbf{2a}}}{C^\prime_{\mathbf{2a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{2a}}}{B^{\prime\prime}_{\mathbf{2a}}}{C^{\prime\prime}_{\mathbf{2a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{2a}}}{B^{\prime\prime\prime}_{\mathbf{2a}}}{C^{\prime\prime\prime}_{\mathbf{2a}}}\)
\(\triangle{A^*_{\mathbf{2a}}}{B^*_{\mathbf{2a}}}{C^*_{\mathbf{2a}}}\)
2a (031)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{2a}}}}&{}\approx{}&0.490909090909&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{2a}}}}&{}\approx{}&4.583333333333&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{2a}}}}&{}\approx{}&0.081481481481&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.750000000000&{}:{}&0.811688311688&{}:{}&0.938311688312&,\\ A^\prime_{\mathbf{2a}}&{}\approx{}&0.140909090909&{}:{}&0.398465171192&{}:{}&0.460625737898&,\\B^\prime_{\mathbf{2a}}&{}\approx{}&-3.437500000000&{}:{}&0.136904761905&{}:{}&4.300595238095&,\\C^\prime_{\mathbf{2a}}&{}\approx{}&-0.061111111111&{}:{}&0.066137566138&{}:{}&0.994973544974&. \end{alignedat} \]
2a (031)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{2a}}&{}\approx{}&-0.034222708945&{}:{}&0.062208884749&{}:{}&0.972013824197&,\\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.120087336245&{}:{}&0.067373674361&{}:{}&1.052713661884&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.032352941176&{}:{}&0.113445378151&{}:{}&0.918907563025&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.205285159749&{}:{}&0.373160431685&{}:{}&0.832124728064&. \end{alignedat} \]
2a (031)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{2a}}&{}\approx{}&0.022911739411&{}:{}&0.002384279878&{}:{}&0.974703980711&,\\ A^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.065268987342&{}:{}&0.002599457505&{}:{}&1.062669529837&,\\B^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.021565899317&{}:{}&0.060984424126&{}:{}&0.917449676557&,\\C^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.228674892704&{}:{}&0.023796750460&{}:{}&0.747528356836&. \end{alignedat} \]
2a (031)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{2a}}&{}\approx{}&-0.083491702976&{}:{}&0.113797692990&{}:{}&0.969694009986&,\\ A^\prime_{\mathbf{2a}}&{}\approx{}&0.140909090909&{}:{}&0.398465171192&{}:{}&0.460625737898&,\\B^\prime_{\mathbf{2a}}&{}\approx{}&-3.437500000000&{}:{}&0.136904761905&{}:{}&4.300595238095&,\\C^\prime_{\mathbf{2a}}&{}\approx{}&-0.061111111111&{}:{}&0.066137566138&{}:{}&0.994973544974&,\\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.120087336245&{}:{}&0.067373674361&{}:{}&1.052713661884&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.032352941176&{}:{}&0.113445378151&{}:{}&0.918907563025&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.205285159749&{}:{}&0.373160431685&{}:{}&0.832124728064&, \end{alignedat} \]
2a (031)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{2a}}&{}\approx{}&0.136986301370&{}:{}&-0.369863013699&{}:{}&1.232876712329&,\\ A^*_{\mathbf{2a}}&{}\approx{}&0.000000000000&{}:{}&-0.428571428571&{}:{}&1.428571428571&,\\B^*_{\mathbf{2a}}&{}\approx{}&0.100000000000&{}:{}&0.000000000000&{}:{}&0.900000000000&,\\C^*_{\mathbf{2a}}&{}\approx{}&-0.588235294118&{}:{}&1.588235294118&{}:{}&0.000000000000&. \end{alignedat} \]
2a (031)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{2a}}&{}\approx{}&-0.157142857143&{}:{}&0.220408163265&{}:{}&0.936734693878&,\\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.120087336245&{}:{}&0.067373674361&{}:{}&1.052713661884&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.032352941176&{}:{}&0.113445378151&{}:{}&0.918907563025&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.205285159749&{}:{}&0.373160431685&{}:{}&0.832124728064&,\\ A^*_{\mathbf{2a}}&{}\approx{}&0.000000000000&{}:{}&-0.428571428571&{}:{}&1.428571428571&,\\B^*_{\mathbf{2a}}&{}\approx{}&0.100000000000&{}:{}&0.000000000000&{}:{}&0.900000000000&,\\C^*_{\mathbf{2a}}&{}\approx{}&-0.588235294118&{}:{}&1.588235294118&{}:{}&0.000000000000&. \end{alignedat} \]
2a (031)