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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4a(211)

Malfatti circles

4a (211)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.750000000000&{}:{}&0.811688311688&{}:{}&0.938311688312&, \\ P_{\mathbf{4a}}&{}\approx{}&1.115409836066&{}:{}&-0.038152011923&{}:{}&-0.077257824143&, \\ P^-_{\mathbf{4a}}&{}\approx{}&0.652045342533&{}:{}&0.172946816614&{}:{}&0.175007840853&, \\ P^+_{\mathbf{4a}}&{}\approx{}&2.036238981391&{}:{}&-0.457661828867&{}:{}&-0.578577152524&, \\ Q_{\mathbf{4a}}&{}\approx{}&5.400000000000&{}:{}&-2.000000000000&{}:{}&-2.400000000000&, \\ I^\prime_{\mathbf{4a}}&{}\approx{}&1.948453608247&{}:{}&-0.374882849110&{}:{}&-0.573570759138&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{4a}}\)
\(P^-_{\mathbf{4a}}\)
\(P^+_{\mathbf{4a}}\)
\(Q_{\mathbf{4a}}\)
\(I^\prime_{\mathbf{4a}}\)
4a (211)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4a}}&{}\approx{}&3.000000000000&{}:{}&-0.927643784787&{}:{}&-1.072356215213&,\\B^\prime_{\mathbf{4a}}&{}\approx{}&1.476562500000&{}:{}&1.370738636364&{}:{}&-1.847301136364&,\\C^\prime_{\mathbf{4a}}&{}\approx{}&1.166666666667&{}:{}&-1.262626262626&{}:{}&1.095959595960&, \\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.303448275862&{}:{}&-0.100313479624&{}:{}&-0.203134796238&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.223529411765&{}:{}&-1.069518716578&{}:{}&-0.154010695187&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.440459110473&{}:{}&-0.083474631538&{}:{}&-1.356984478936&, \\ A^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&0.348708487085&{}:{}&0.323716873532&{}:{}&0.327574639383&,\\B^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.042553191489&{}:{}&-0.322372662798&{}:{}&0.279819471309&,\\C^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.101582014988&{}:{}&0.292180758459&{}:{}&-0.393762773446&, \\ A^*_{\mathbf{4a}}&{}\approx{}&0.000000000000&{}:{}&0.454545454545&{}:{}&0.545454545455&,\\B^*_{\mathbf{4a}}&{}\approx{}&1.800000000000&{}:{}&0.000000000000&{}:{}&-0.800000000000&,\\C^*_{\mathbf{4a}}&{}\approx{}&1.588235294118&{}:{}&-0.588235294118&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4a}}}{B^\prime_{\mathbf{4a}}}{C^\prime_{\mathbf{4a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4a}}}{B^{\prime\prime}_{\mathbf{4a}}}{C^{\prime\prime}_{\mathbf{4a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4a}}}{B^{\prime\prime\prime}_{\mathbf{4a}}}{C^{\prime\prime\prime}_{\mathbf{4a}}}\)
\(\triangle{A^*_{\mathbf{4a}}}{B^*_{\mathbf{4a}}}{C^*_{\mathbf{4a}}}\)
4a (211)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4a}}}}&{}\approx{}&-1.142857142857&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4a}}}}&{}\approx{}&-1.968750000000&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4a}}}}&{}\approx{}&-1.555555555556&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.750000000000&{}:{}&0.811688311688&{}:{}&0.938311688312&,\\ A^\prime_{\mathbf{4a}}&{}\approx{}&3.000000000000&{}:{}&-0.927643784787&{}:{}&-1.072356215213&,\\B^\prime_{\mathbf{4a}}&{}\approx{}&1.476562500000&{}:{}&1.370738636364&{}:{}&-1.847301136364&,\\C^\prime_{\mathbf{4a}}&{}\approx{}&1.166666666667&{}:{}&-1.262626262626&{}:{}&1.095959595960&. \end{alignedat} \]
4a (211)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4a}}&{}\approx{}&1.115409836066&{}:{}&-0.038152011923&{}:{}&-0.077257824143&,\\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.303448275862&{}:{}&-0.100313479624&{}:{}&-0.203134796238&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.223529411765&{}:{}&-1.069518716578&{}:{}&-0.154010695187&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.440459110473&{}:{}&-0.083474631538&{}:{}&-1.356984478936&. \end{alignedat} \]
4a (211)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4a}}&{}\approx{}&0.652045342533&{}:{}&0.172946816614&{}:{}&0.175007840853&,\\ A^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&0.348708487085&{}:{}&0.323716873532&{}:{}&0.327574639383&,\\B^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.042553191489&{}:{}&-0.322372662798&{}:{}&0.279819471309&,\\C^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.101582014988&{}:{}&0.292180758459&{}:{}&-0.393762773446&. \end{alignedat} \]
4a (211)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4a}}&{}\approx{}&2.036238981391&{}:{}&-0.457661828867&{}:{}&-0.578577152524&,\\ A^\prime_{\mathbf{4a}}&{}\approx{}&3.000000000000&{}:{}&-0.927643784787&{}:{}&-1.072356215213&,\\B^\prime_{\mathbf{4a}}&{}\approx{}&1.476562500000&{}:{}&1.370738636364&{}:{}&-1.847301136364&,\\C^\prime_{\mathbf{4a}}&{}\approx{}&1.166666666667&{}:{}&-1.262626262626&{}:{}&1.095959595960&,\\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.303448275862&{}:{}&-0.100313479624&{}:{}&-0.203134796238&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.223529411765&{}:{}&-1.069518716578&{}:{}&-0.154010695187&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.440459110473&{}:{}&-0.083474631538&{}:{}&-1.356984478936&, \end{alignedat} \]
4a (211)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4a}}&{}\approx{}&5.400000000000&{}:{}&-2.000000000000&{}:{}&-2.400000000000&,\\ A^*_{\mathbf{4a}}&{}\approx{}&0.000000000000&{}:{}&0.454545454545&{}:{}&0.545454545455&,\\B^*_{\mathbf{4a}}&{}\approx{}&1.800000000000&{}:{}&0.000000000000&{}:{}&-0.800000000000&,\\C^*_{\mathbf{4a}}&{}\approx{}&1.588235294118&{}:{}&-0.588235294118&{}:{}&0.000000000000&. \end{alignedat} \]
4a (211)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4a}}&{}\approx{}&1.948453608247&{}:{}&-0.374882849110&{}:{}&-0.573570759138&,\\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.303448275862&{}:{}&-0.100313479624&{}:{}&-0.203134796238&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.223529411765&{}:{}&-1.069518716578&{}:{}&-0.154010695187&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.440459110473&{}:{}&-0.083474631538&{}:{}&-1.356984478936&,\\ A^*_{\mathbf{4a}}&{}\approx{}&0.000000000000&{}:{}&0.454545454545&{}:{}&0.545454545455&,\\B^*_{\mathbf{4a}}&{}\approx{}&1.800000000000&{}:{}&0.000000000000&{}:{}&-0.800000000000&,\\C^*_{\mathbf{4a}}&{}\approx{}&1.588235294118&{}:{}&-0.588235294118&{}:{}&0.000000000000&. \end{alignedat} \]
4a (211)