Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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4(200)

Malfatti circles

4 (200)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.200000000000&{}:{}&0.333333333333&{}:{}&0.466666666667&, \\ P_{\mathbf{4}}&{}\approx{}&0.999520436600&{}:{}&0.000142996715&{}:{}&0.000336566685&, \\ P^-_{\mathbf{4}}&{}\approx{}&1.012053812179&{}:{}&-0.005080133848&{}:{}&-0.006973678331&, \\ P^+_{\mathbf{4}}&{}\approx{}&0.987368065008&{}:{}&0.005207348539&{}:{}&0.007424586453&, \\ Q_{\mathbf{4}}&{}\approx{}&1.076539617477&{}:{}&-0.017275612770&{}:{}&-0.059264004707&, \\ I^\prime_{\mathbf{4}}&{}\approx{}&0.958339177159&{}:{}&0.014798265296&{}:{}&0.026862557545&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{4}}\)
\(P^-_{\mathbf{4}}\)
\(P^+_{\mathbf{4}}\)
\(Q_{\mathbf{4}}\)
\(I^\prime_{\mathbf{4}}\)
4 (200)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4}}&{}\approx{}&0.987760142421&{}:{}&0.005099940658&{}:{}&0.007139916921&,\\B^\prime_{\mathbf{4}}&{}\approx{}&0.583774096392&{}:{}&-0.945913654641&{}:{}&1.362139558248&,\\C^\prime_{\mathbf{4}}&{}\approx{}&2.260569957535&{}:{}&3.767616595892&{}:{}&-5.028186553427&, \\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.927920222920&{}:{}&0.021492823146&{}:{}&0.050586953934&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.989480720806&{}:{}&0.010186093164&{}:{}&0.000333186030&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.985653613507&{}:{}&0.000141012854&{}:{}&0.014205373639&, \\ A^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&-0.351538047846&{}:{}&0.569611844100&{}:{}&0.781926203747&,\\B^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&1.001730472737&{}:{}&0.005172071383&{}:{}&-0.006902544121&,\\C^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&0.997796104673&{}:{}&-0.005008565457&{}:{}&0.007212460784&, \\ A^*_{\mathbf{4}}&{}\approx{}&0.000000000000&{}:{}&0.225708114823&{}:{}&0.774291885177&,\\B^*_{\mathbf{4}}&{}\approx{}&1.058257569496&{}:{}&0.000000000000&{}:{}&-0.058257569496&,\\C^*_{\mathbf{4}}&{}\approx{}&1.016309071859&{}:{}&-0.016309071859&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4}}}{B^\prime_{\mathbf{4}}}{C^\prime_{\mathbf{4}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4}}}{B^{\prime\prime}_{\mathbf{4}}}{C^{\prime\prime}_{\mathbf{4}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4}}}{B^{\prime\prime\prime}_{\mathbf{4}}}{C^{\prime\prime\prime}_{\mathbf{4}}}\)
\(\triangle{A^*_{\mathbf{4}}}{B^*_{\mathbf{4}}}{C^*_{\mathbf{4}}}\)
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Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4}}}}&{}\approx{}&0.015299821974&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4}}}}&{}\approx{}&2.918870481961&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4}}}}&{}\approx{}&11.302849787676&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.200000000000&{}:{}&0.333333333333&{}:{}&0.466666666667&,\\ A^\prime_{\mathbf{4}}&{}\approx{}&0.987760142421&{}:{}&0.005099940658&{}:{}&0.007139916921&,\\B^\prime_{\mathbf{4}}&{}\approx{}&0.583774096392&{}:{}&-0.945913654641&{}:{}&1.362139558248&,\\C^\prime_{\mathbf{4}}&{}\approx{}&2.260569957535&{}:{}&3.767616595892&{}:{}&-5.028186553427&. \end{alignedat} \]
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First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4}}&{}\approx{}&0.999520436600&{}:{}&0.000142996715&{}:{}&0.000336566685&,\\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.927920222920&{}:{}&0.021492823146&{}:{}&0.050586953934&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.989480720806&{}:{}&0.010186093164&{}:{}&0.000333186030&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.985653613507&{}:{}&0.000141012854&{}:{}&0.014205373639&. \end{alignedat} \]
4 (200)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4}}&{}\approx{}&1.012053812179&{}:{}&-0.005080133848&{}:{}&-0.006973678331&,\\ A^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&-0.351538047846&{}:{}&0.569611844100&{}:{}&0.781926203747&,\\B^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&1.001730472737&{}:{}&0.005172071383&{}:{}&-0.006902544121&,\\C^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&0.997796104673&{}:{}&-0.005008565457&{}:{}&0.007212460784&. \end{alignedat} \]
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Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4}}&{}\approx{}&0.987368065008&{}:{}&0.005207348539&{}:{}&0.007424586453&,\\ A^\prime_{\mathbf{4}}&{}\approx{}&0.987760142421&{}:{}&0.005099940658&{}:{}&0.007139916921&,\\B^\prime_{\mathbf{4}}&{}\approx{}&0.583774096392&{}:{}&-0.945913654641&{}:{}&1.362139558248&,\\C^\prime_{\mathbf{4}}&{}\approx{}&2.260569957535&{}:{}&3.767616595892&{}:{}&-5.028186553427&,\\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.927920222920&{}:{}&0.021492823146&{}:{}&0.050586953934&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.989480720806&{}:{}&0.010186093164&{}:{}&0.000333186030&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.985653613507&{}:{}&0.000141012854&{}:{}&0.014205373639&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4}}&{}\approx{}&1.076539617477&{}:{}&-0.017275612770&{}:{}&-0.059264004707&,\\ A^*_{\mathbf{4}}&{}\approx{}&0.000000000000&{}:{}&0.225708114823&{}:{}&0.774291885177&,\\B^*_{\mathbf{4}}&{}\approx{}&1.058257569496&{}:{}&0.000000000000&{}:{}&-0.058257569496&,\\C^*_{\mathbf{4}}&{}\approx{}&1.016309071859&{}:{}&-0.016309071859&{}:{}&0.000000000000&. \end{alignedat} \]
4 (200)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4}}&{}\approx{}&0.958339177159&{}:{}&0.014798265296&{}:{}&0.026862557545&,\\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.927920222920&{}:{}&0.021492823146&{}:{}&0.050586953934&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.989480720806&{}:{}&0.010186093164&{}:{}&0.000333186030&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.985653613507&{}:{}&0.000141012854&{}:{}&0.014205373639&,\\ A^*_{\mathbf{4}}&{}\approx{}&0.000000000000&{}:{}&0.225708114823&{}:{}&0.774291885177&,\\B^*_{\mathbf{4}}&{}\approx{}&1.058257569496&{}:{}&0.000000000000&{}:{}&-0.058257569496&,\\C^*_{\mathbf{4}}&{}\approx{}&1.016309071859&{}:{}&-0.016309071859&{}:{}&0.000000000000&. \end{alignedat} \]
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