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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2 (020)

Malfatti circles

2 (020)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.300000000000&{}:{}&0.324675324675&{}:{}&0.375324675325&, \\ P_{\mathbf{2}}&{}\approx{}&0.003491271820&{}:{}&0.991611879392&{}:{}&0.004896848787&, \\ P^-_{\mathbf{2}}&{}\approx{}&-0.012894736842&{}:{}&1.028468899522&{}:{}&-0.015574162679&, \\ P^+_{\mathbf{2}}&{}\approx{}&0.018246445498&{}:{}&0.958423093494&{}:{}&0.023330461008&, \\ Q_{\mathbf{2}}&{}\approx{}&-0.064516129032&{}:{}&1.161290322581&{}:{}&-0.096774193548&, \\ I^\prime_{\mathbf{2}}&{}\approx{}&0.050359712230&{}:{}&0.882930019621&{}:{}&0.066710268149&, \end{alignedat} \]
\(I\) Incenter
\(P_{\mathbf{2}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{2}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{2}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{2}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{2}}\) Radical center of the Malfatti circles
2 (020)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{2}}&{}\approx{}&-0.500000000000&{}:{}&0.695732838590&{}:{}&0.804267161410&,\\B^\prime_{\mathbf{2}}&{}\approx{}&0.015555555556&{}:{}&0.964983164983&{}:{}&0.019461279461&,\\C^\prime_{\mathbf{2}}&{}\approx{}&0.875000000000&{}:{}&0.946969696970&{}:{}&-0.821969696970&, \\ A^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.030567685590&{}:{}&0.964668519254&{}:{}&0.004763795157&,\\B^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.082352941176&{}:{}&0.802139037433&{}:{}&0.115508021390&,\\C^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.003375120540&{}:{}&0.958621898834&{}:{}&0.038002980626&, \\ A^{\prime\prime\prime}_{\mathbf{2}}&{}\approx{}&0.016104294479&{}:{}&0.999023982153&{}:{}&-0.015128276631&,\\B^{\prime\prime\prime}_{\mathbf{2}}&{}\approx{}&1.225000000000&{}:{}&-1.704545454545&{}:{}&1.479545454545&,\\C^{\prime\prime\prime}_{\mathbf{2}}&{}\approx{}&-0.012442864398&{}:{}&0.992428089940&{}:{}&0.020014774459&, \\ A^*_{\mathbf{2}}&{}\approx{}&0.000000000000&{}:{}&1.090909090909&{}:{}&-0.090909090909&,\\B^*_{\mathbf{2}}&{}\approx{}&0.400000000000&{}:{}&0.000000000000&{}:{}&0.600000000000&,\\C^*_{\mathbf{2}}&{}\approx{}&-0.058823529412&{}:{}&1.058823529412&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{2}}}{B^\prime_{\mathbf{2}}}{C^\prime_{\mathbf{2}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{2}}}{B^{\prime\prime}_{\mathbf{2}}}{C^{\prime\prime}_{\mathbf{2}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{2}}}{B^{\prime\prime\prime}_{\mathbf{2}}}{C^{\prime\prime\prime}_{\mathbf{2}}}\)
\(\triangle{A^*_{\mathbf{2}}}{B^*_{\mathbf{2}}}{C^*_{\mathbf{2}}}\)
2 (020)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{2}}}}&{}\approx{}&2.142857142857&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{2}}}}&{}\approx{}&0.051851851852&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{2}}}}&{}\approx{}&2.916666666667&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.300000000000&{}:{}&0.324675324675&{}:{}&0.375324675325&,\\ A^\prime_{\mathbf{2}}&{}\approx{}&-0.500000000000&{}:{}&0.695732838590&{}:{}&0.804267161410&,\\B^\prime_{\mathbf{2}}&{}\approx{}&0.015555555556&{}:{}&0.964983164983&{}:{}&0.019461279461&,\\C^\prime_{\mathbf{2}}&{}\approx{}&0.875000000000&{}:{}&0.946969696970&{}:{}&-0.821969696970&. \end{alignedat} \]
2 (020)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{2}}&{}\approx{}&0.003491271820&{}:{}&0.991611879392&{}:{}&0.004896848787&,\\ A^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.030567685590&{}:{}&0.964668519254&{}:{}&0.004763795157&,\\B^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.082352941176&{}:{}&0.802139037433&{}:{}&0.115508021390&,\\C^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.003375120540&{}:{}&0.958621898834&{}:{}&0.038002980626&. \end{alignedat} \]
2 (020)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{2}}&{}\approx{}&-0.012894736842&{}:{}&1.028468899522&{}:{}&-0.015574162679&,\\ A^{\prime\prime\prime}_{\mathbf{2}}&{}\approx{}&0.016104294479&{}:{}&0.999023982153&{}:{}&-0.015128276631&,\\B^{\prime\prime\prime}_{\mathbf{2}}&{}\approx{}&1.225000000000&{}:{}&-1.704545454545&{}:{}&1.479545454545&,\\C^{\prime\prime\prime}_{\mathbf{2}}&{}\approx{}&-0.012442864398&{}:{}&0.992428089940&{}:{}&0.020014774459&. \end{alignedat} \]
2 (020)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{2}}&{}\approx{}&0.018246445498&{}:{}&0.958423093494&{}:{}&0.023330461008&,\\ A^\prime_{\mathbf{2}}&{}\approx{}&-0.500000000000&{}:{}&0.695732838590&{}:{}&0.804267161410&,\\B^\prime_{\mathbf{2}}&{}\approx{}&0.015555555556&{}:{}&0.964983164983&{}:{}&0.019461279461&,\\C^\prime_{\mathbf{2}}&{}\approx{}&0.875000000000&{}:{}&0.946969696970&{}:{}&-0.821969696970&,\\ A^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.030567685590&{}:{}&0.964668519254&{}:{}&0.004763795157&,\\B^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.082352941176&{}:{}&0.802139037433&{}:{}&0.115508021390&,\\C^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.003375120540&{}:{}&0.958621898834&{}:{}&0.038002980626&, \end{alignedat} \]
2 (020)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{2}}&{}\approx{}&-0.064516129032&{}:{}&1.161290322581&{}:{}&-0.096774193548&,\\ A^*_{\mathbf{2}}&{}\approx{}&0.000000000000&{}:{}&1.090909090909&{}:{}&-0.090909090909&,\\B^*_{\mathbf{2}}&{}\approx{}&0.400000000000&{}:{}&0.000000000000&{}:{}&0.600000000000&,\\C^*_{\mathbf{2}}&{}\approx{}&-0.058823529412&{}:{}&1.058823529412&{}:{}&0.000000000000&. \end{alignedat} \]
2 (020)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{2}}&{}\approx{}&0.050359712230&{}:{}&0.882930019621&{}:{}&0.066710268149&,\\ A^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.030567685590&{}:{}&0.964668519254&{}:{}&0.004763795157&,\\B^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.082352941176&{}:{}&0.802139037433&{}:{}&0.115508021390&,\\C^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.003375120540&{}:{}&0.958621898834&{}:{}&0.038002980626&,\\ A^*_{\mathbf{2}}&{}\approx{}&0.000000000000&{}:{}&1.090909090909&{}:{}&-0.090909090909&,\\B^*_{\mathbf{2}}&{}\approx{}&0.400000000000&{}:{}&0.000000000000&{}:{}&0.600000000000&,\\C^*_{\mathbf{2}}&{}\approx{}&-0.058823529412&{}:{}&1.058823529412&{}:{}&0.000000000000&. \end{alignedat} \]
2 (020)