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”.

\(a:b:c=231:250:289\).


[Top] > Martin's solution > 1b (103)

1b(103)

Malfatti circles

1b (103)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.855555555556&{}:{}&-0.925925925926&{}:{}&1.070370370370&, \\ P_{\mathbf{1b}}&{}\approx{}&0.942594142259&{}:{}&-0.095676429568&{}:{}&0.153082287308&, \\ P^-_{\mathbf{1b}}&{}\approx{}&0.952853133770&{}:{}&0.002182725288&{}:{}&0.044964140942&, \\ P^+_{\mathbf{1b}}&{}\approx{}&0.934292202877&{}:{}&-0.174867524603&{}:{}&0.240575321726&, \\ Q_{\mathbf{1b}}&{}\approx{}&1.189189189189&{}:{}&0.189189189189&{}:{}&-0.378378378378&, \\ I^\prime_{\mathbf{1b}}&{}\approx{}&0.893401015228&{}:{}&-0.296108291032&{}:{}&0.402707275804&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{1b}}\)
\(P^-_{\mathbf{1b}}\)
\(P^+_{\mathbf{1b}}\)
\(Q_{\mathbf{1b}}\)
\(I^\prime_{\mathbf{1b}}\)
1b (103)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1b}}&{}\approx{}&0.982445987654&{}:{}&-0.112525720165&{}:{}&0.130079732510&,\\B^\prime_{\mathbf{1b}}&{}\approx{}&0.440000000000&{}:{}&0.009523809524&{}:{}&0.550476190476&,\\C^\prime_{\mathbf{1b}}&{}\approx{}&2.750000000000&{}:{}&-2.976190476190&{}:{}&1.226190476190&, \\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.758620689655&{}:{}&-0.402298850575&{}:{}&0.643678160920&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.046840148699&{}:{}&-0.216852540273&{}:{}&0.170012391574&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.878764237791&{}:{}&-0.089197482707&{}:{}&0.210433244916&, \\ A^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.681415929204&{}:{}&0.014749262537&{}:{}&0.303834808260&,\\B^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.072210526316&{}:{}&-0.122807017544&{}:{}&0.050596491228&,\\C^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.881294341582&{}:{}&0.002018803715&{}:{}&0.116686854704&, \\ A^*_{\mathbf{1b}}&{}\approx{}&0.000000000000&{}:{}&-1.000000000000&{}:{}&2.000000000000&,\\B^*_{\mathbf{1b}}&{}\approx{}&1.466666666667&{}:{}&0.000000000000&{}:{}&-0.466666666667&,\\C^*_{\mathbf{1b}}&{}\approx{}&0.862745098039&{}:{}&0.137254901961&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1b}}}{B^\prime_{\mathbf{1b}}}{C^\prime_{\mathbf{1b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1b}}}{B^{\prime\prime}_{\mathbf{1b}}}{C^{\prime\prime}_{\mathbf{1b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1b}}}{B^{\prime\prime\prime}_{\mathbf{1b}}}{C^{\prime\prime\prime}_{\mathbf{1b}}}\)
\(\triangle{A^*_{\mathbf{1b}}}{B^*_{\mathbf{1b}}}{C^*_{\mathbf{1b}}}\)
1b (103)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1b}}}}&{}\approx{}&0.121527777778&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1b}}}}&{}\approx{}&0.514285714286&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1b}}}}&{}\approx{}&3.214285714286&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.855555555556&{}:{}&-0.925925925926&{}:{}&1.070370370370&,\\ A^\prime_{\mathbf{1b}}&{}\approx{}&0.982445987654&{}:{}&-0.112525720165&{}:{}&0.130079732510&,\\B^\prime_{\mathbf{1b}}&{}\approx{}&0.440000000000&{}:{}&0.009523809524&{}:{}&0.550476190476&,\\C^\prime_{\mathbf{1b}}&{}\approx{}&2.750000000000&{}:{}&-2.976190476190&{}:{}&1.226190476190&. \end{alignedat} \]
1b (103)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1b}}&{}\approx{}&0.942594142259&{}:{}&-0.095676429568&{}:{}&0.153082287308&,\\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.758620689655&{}:{}&-0.402298850575&{}:{}&0.643678160920&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.046840148699&{}:{}&-0.216852540273&{}:{}&0.170012391574&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.878764237791&{}:{}&-0.089197482707&{}:{}&0.210433244916&. \end{alignedat} \]
1b (103)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1b}}&{}\approx{}&0.952853133770&{}:{}&0.002182725288&{}:{}&0.044964140942&,\\ A^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.681415929204&{}:{}&0.014749262537&{}:{}&0.303834808260&,\\B^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.072210526316&{}:{}&-0.122807017544&{}:{}&0.050596491228&,\\C^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.881294341582&{}:{}&0.002018803715&{}:{}&0.116686854704&. \end{alignedat} \]
1b (103)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1b}}&{}\approx{}&0.934292202877&{}:{}&-0.174867524603&{}:{}&0.240575321726&,\\ A^\prime_{\mathbf{1b}}&{}\approx{}&0.982445987654&{}:{}&-0.112525720165&{}:{}&0.130079732510&,\\B^\prime_{\mathbf{1b}}&{}\approx{}&0.440000000000&{}:{}&0.009523809524&{}:{}&0.550476190476&,\\C^\prime_{\mathbf{1b}}&{}\approx{}&2.750000000000&{}:{}&-2.976190476190&{}:{}&1.226190476190&,\\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.758620689655&{}:{}&-0.402298850575&{}:{}&0.643678160920&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.046840148699&{}:{}&-0.216852540273&{}:{}&0.170012391574&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.878764237791&{}:{}&-0.089197482707&{}:{}&0.210433244916&, \end{alignedat} \]
1b (103)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1b}}&{}\approx{}&1.189189189189&{}:{}&0.189189189189&{}:{}&-0.378378378378&,\\ A^*_{\mathbf{1b}}&{}\approx{}&0.000000000000&{}:{}&-1.000000000000&{}:{}&2.000000000000&,\\B^*_{\mathbf{1b}}&{}\approx{}&1.466666666667&{}:{}&0.000000000000&{}:{}&-0.466666666667&,\\C^*_{\mathbf{1b}}&{}\approx{}&0.862745098039&{}:{}&0.137254901961&{}:{}&0.000000000000&. \end{alignedat} \]
1b (103)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1b}}&{}\approx{}&0.893401015228&{}:{}&-0.296108291032&{}:{}&0.402707275804&,\\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.758620689655&{}:{}&-0.402298850575&{}:{}&0.643678160920&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.046840148699&{}:{}&-0.216852540273&{}:{}&0.170012391574&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.878764237791&{}:{}&-0.089197482707&{}:{}&0.210433244916&,\\ A^*_{\mathbf{1b}}&{}\approx{}&0.000000000000&{}:{}&-1.000000000000&{}:{}&2.000000000000&,\\B^*_{\mathbf{1b}}&{}\approx{}&1.466666666667&{}:{}&0.000000000000&{}:{}&-0.466666666667&,\\C^*_{\mathbf{1b}}&{}\approx{}&0.862745098039&{}:{}&0.137254901961&{}:{}&0.000000000000&. \end{alignedat} \]
1b (103)