Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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2a(031)

Malfatti circles

2a (031)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.333333333333&{}:{}&0.555555555556&{}:{}&0.777777777778&, \\ P_{\mathbf{2a}}&{}\approx{}&-0.001947894931&{}:{}&0.007238645105&{}:{}&0.994709249826&, \\ P^-_{\mathbf{2a}}&{}\approx{}&0.009207054840&{}:{}&-0.011218555743&{}:{}&1.002011500903&, \\ P^+_{\mathbf{2a}}&{}\approx{}&-0.012399228392&{}:{}&0.024531628469&{}:{}&0.987867599923&, \\ Q_{\mathbf{2a}}&{}\approx{}&0.030604998768&{}:{}&-0.228498774114&{}:{}&1.197893775347&, \\ I^\prime_{\mathbf{2a}}&{}\approx{}&-0.027772123623&{}:{}&0.069116118911&{}:{}&0.958656004711&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{2a}}\)
\(P^-_{\mathbf{2a}}\)
\(P^+_{\mathbf{2a}}\)
\(Q_{\mathbf{2a}}\)
\(I^\prime_{\mathbf{2a}}\)
2a (031)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{2a}}&{}\approx{}&0.170220055912&{}:{}&0.345741643370&{}:{}&0.484038300718&,\\B^\prime_{\mathbf{2a}}&{}\approx{}&-2.998426136278&{}:{}&-2.997901515037&{}:{}&6.996327651315&,\\C^\prime_{\mathbf{2a}}&{}\approx{}&-0.011115016909&{}:{}&0.018525028182&{}:{}&0.992589988727&, \\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.022147932531&{}:{}&0.007384581739&{}:{}&1.014763350791&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.001893099021&{}:{}&0.035165850665&{}:{}&0.966727248356&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.034816185400&{}:{}&0.129381726893&{}:{}&0.905434458508&, \\ A^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.011454506729&{}:{}&-0.011452502585&{}:{}&1.022907009314&,\\B^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.008939587559&{}:{}&0.018157600039&{}:{}&0.972902812402&,\\C^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.380933608633&{}:{}&-0.464157648368&{}:{}&1.083224039735&, \\ A^*_{\mathbf{2a}}&{}\approx{}&0.000000000000&{}:{}&-0.235712762933&{}:{}&1.235712762933&,\\B^*_{\mathbf{2a}}&{}\approx{}&0.024912518769&{}:{}&0.000000000000&{}:{}&0.975087481231&,\\C^*_{\mathbf{2a}}&{}\approx{}&-0.154653670708&{}:{}&1.154653670708&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{2a}}}{B^\prime_{\mathbf{2a}}}{C^\prime_{\mathbf{2a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{2a}}}{B^{\prime\prime}_{\mathbf{2a}}}{C^{\prime\prime}_{\mathbf{2a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{2a}}}{B^{\prime\prime\prime}_{\mathbf{2a}}}{C^{\prime\prime\prime}_{\mathbf{2a}}}\)
\(\triangle{A^*_{\mathbf{2a}}}{B^*_{\mathbf{2a}}}{C^*_{\mathbf{2a}}}\)
2a (031)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{2a}}}}&{}\approx{}&0.622334958066&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{2a}}}}&{}\approx{}&8.995278408834&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{2a}}}}&{}\approx{}&0.033345050727&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.333333333333&{}:{}&0.555555555556&{}:{}&0.777777777778&,\\ A^\prime_{\mathbf{2a}}&{}\approx{}&0.170220055912&{}:{}&0.345741643370&{}:{}&0.484038300718&,\\B^\prime_{\mathbf{2a}}&{}\approx{}&-2.998426136278&{}:{}&-2.997901515037&{}:{}&6.996327651315&,\\C^\prime_{\mathbf{2a}}&{}\approx{}&-0.011115016909&{}:{}&0.018525028182&{}:{}&0.992589988727&. \end{alignedat} \]
2a (031)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{2a}}&{}\approx{}&-0.001947894931&{}:{}&0.007238645105&{}:{}&0.994709249826&,\\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.022147932531&{}:{}&0.007384581739&{}:{}&1.014763350791&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.001893099021&{}:{}&0.035165850665&{}:{}&0.966727248356&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.034816185400&{}:{}&0.129381726893&{}:{}&0.905434458508&. \end{alignedat} \]
2a (031)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{2a}}&{}\approx{}&0.009207054840&{}:{}&-0.011218555743&{}:{}&1.002011500903&,\\ A^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.011454506729&{}:{}&-0.011452502585&{}:{}&1.022907009314&,\\B^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.008939587559&{}:{}&0.018157600039&{}:{}&0.972902812402&,\\C^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.380933608633&{}:{}&-0.464157648368&{}:{}&1.083224039735&. \end{alignedat} \]
2a (031)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{2a}}&{}\approx{}&-0.012399228392&{}:{}&0.024531628469&{}:{}&0.987867599923&,\\ A^\prime_{\mathbf{2a}}&{}\approx{}&0.170220055912&{}:{}&0.345741643370&{}:{}&0.484038300718&,\\B^\prime_{\mathbf{2a}}&{}\approx{}&-2.998426136278&{}:{}&-2.997901515037&{}:{}&6.996327651315&,\\C^\prime_{\mathbf{2a}}&{}\approx{}&-0.011115016909&{}:{}&0.018525028182&{}:{}&0.992589988727&,\\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.022147932531&{}:{}&0.007384581739&{}:{}&1.014763350791&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.001893099021&{}:{}&0.035165850665&{}:{}&0.966727248356&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.034816185400&{}:{}&0.129381726893&{}:{}&0.905434458508&, \end{alignedat} \]
2a (031)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{2a}}&{}\approx{}&0.030604998768&{}:{}&-0.228498774114&{}:{}&1.197893775347&,\\ A^*_{\mathbf{2a}}&{}\approx{}&0.000000000000&{}:{}&-0.235712762933&{}:{}&1.235712762933&,\\B^*_{\mathbf{2a}}&{}\approx{}&0.024912518769&{}:{}&0.000000000000&{}:{}&0.975087481231&,\\C^*_{\mathbf{2a}}&{}\approx{}&-0.154653670708&{}:{}&1.154653670708&{}:{}&0.000000000000&. \end{alignedat} \]
2a (031)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{2a}}&{}\approx{}&-0.027772123623&{}:{}&0.069116118911&{}:{}&0.958656004711&,\\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.022147932531&{}:{}&0.007384581739&{}:{}&1.014763350791&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.001893099021&{}:{}&0.035165850665&{}:{}&0.966727248356&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.034816185400&{}:{}&0.129381726893&{}:{}&0.905434458508&,\\ A^*_{\mathbf{2a}}&{}\approx{}&0.000000000000&{}:{}&-0.235712762933&{}:{}&1.235712762933&,\\B^*_{\mathbf{2a}}&{}\approx{}&0.024912518769&{}:{}&0.000000000000&{}:{}&0.975087481231&,\\C^*_{\mathbf{2a}}&{}\approx{}&-0.154653670708&{}:{}&1.154653670708&{}:{}&0.000000000000&. \end{alignedat} \]
2a (031)