Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).

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1a (013)

Malfatti circles

1a (013)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.333333333333&{}:{}&0.555555555556&{}:{}&0.777777777778&, \\ P_{\mathbf{1a}}&{}\approx{}&-0.062106567306&{}:{}&0.898621547287&{}:{}&0.163485020019&, \\ P^-_{\mathbf{1a}}&{}\approx{}&-0.015328758158&{}:{}&0.957789289916&{}:{}&0.057539468242&, \\ P^+_{\mathbf{1a}}&{}\approx{}&-0.096887287105&{}:{}&0.854628536176&{}:{}&0.242258750929&, \\ Q_{\mathbf{1a}}&{}\approx{}&0.077387626438&{}:{}&1.140083885251&{}:{}&-0.217471511689&, \\ I^\prime_{\mathbf{1a}}&{}\approx{}&-0.156518182688&{}:{}&0.768614941193&{}:{}&0.387903241495&, \end{alignedat} \]
\(I_{\mathbf{a}}\) Incenter
\(P_{\mathbf{1a}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{1a}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{1a}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{1a}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{1a}}\) Radical center of the Malfatti circles
1a (013)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1a}}&{}\approx{}&-0.071420142185&{}:{}&0.446425059244&{}:{}&0.624995082941&,\\B^\prime_{\mathbf{1a}}&{}\approx{}&-0.055575084545&{}:{}&0.925899887274&{}:{}&0.129675197271&,\\C^\prime_{\mathbf{1a}}&{}\approx{}&-0.599685227256&{}:{}&0.999475378759&{}:{}&0.600209848496&, \\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.101723106389&{}:{}&0.932140100646&{}:{}&0.169583005743&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.234523192506&{}:{}&0.617180607251&{}:{}&0.617342585256&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.058297745062&{}:{}&0.843511598585&{}:{}&0.214786146477&, \\ A^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.060019627647&{}:{}&0.999947493168&{}:{}&0.060072134479&,\\B^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.111057032304&{}:{}&0.694182911279&{}:{}&0.416874121026&,\\C^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.014238092635&{}:{}&0.889641058594&{}:{}&0.124597034041&, \\ A^*_{\mathbf{1a}}&{}\approx{}&0.000000000000&{}:{}&1.235712762933&{}:{}&-0.235712762933&,\\B^*_{\mathbf{1a}}&{}\approx{}&-0.552437750421&{}:{}&0.000000000000&{}:{}&1.552437750421&,\\C^*_{\mathbf{1a}}&{}\approx{}&0.063564219528&{}:{}&0.936435780472&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1a}}}{B^\prime_{\mathbf{1a}}}{C^\prime_{\mathbf{1a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1a}}}{B^{\prime\prime}_{\mathbf{1a}}}{C^{\prime\prime}_{\mathbf{1a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1a}}}{B^{\prime\prime\prime}_{\mathbf{1a}}}{C^{\prime\prime\prime}_{\mathbf{1a}}}\)
\(\triangle{A^*_{\mathbf{1a}}}{B^*_{\mathbf{1a}}}{C^*_{\mathbf{1a}}}\)
1a (013)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1a}}}}&{}\approx{}&0.803565106639&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1a}}}}&{}\approx{}&0.166725253634&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1a}}}}&{}\approx{}&1.799055681767&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.333333333333&{}:{}&0.555555555556&{}:{}&0.777777777778&,\\ A^\prime_{\mathbf{1a}}&{}\approx{}&-0.071420142185&{}:{}&0.446425059244&{}:{}&0.624995082941&,\\B^\prime_{\mathbf{1a}}&{}\approx{}&-0.055575084545&{}:{}&0.925899887274&{}:{}&0.129675197271&,\\C^\prime_{\mathbf{1a}}&{}\approx{}&-0.599685227256&{}:{}&0.999475378759&{}:{}&0.600209848496&. \end{alignedat} \]
1a (013)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1a}}&{}\approx{}&-0.062106567306&{}:{}&0.898621547287&{}:{}&0.163485020019&,\\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.101723106389&{}:{}&0.932140100646&{}:{}&0.169583005743&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.234523192506&{}:{}&0.617180607251&{}:{}&0.617342585256&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.058297745062&{}:{}&0.843511598585&{}:{}&0.214786146477&. \end{alignedat} \]
1a (013)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1a}}&{}\approx{}&-0.015328758158&{}:{}&0.957789289916&{}:{}&0.057539468242&,\\ A^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.060019627647&{}:{}&0.999947493168&{}:{}&0.060072134479&,\\B^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.111057032304&{}:{}&0.694182911279&{}:{}&0.416874121026&,\\C^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.014238092635&{}:{}&0.889641058594&{}:{}&0.124597034041&. \end{alignedat} \]
1a (013)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1a}}&{}\approx{}&-0.096887287105&{}:{}&0.854628536176&{}:{}&0.242258750929&,\\ A^\prime_{\mathbf{1a}}&{}\approx{}&-0.071420142185&{}:{}&0.446425059244&{}:{}&0.624995082941&,\\B^\prime_{\mathbf{1a}}&{}\approx{}&-0.055575084545&{}:{}&0.925899887274&{}:{}&0.129675197271&,\\C^\prime_{\mathbf{1a}}&{}\approx{}&-0.599685227256&{}:{}&0.999475378759&{}:{}&0.600209848496&,\\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.101723106389&{}:{}&0.932140100646&{}:{}&0.169583005743&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.234523192506&{}:{}&0.617180607251&{}:{}&0.617342585256&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.058297745062&{}:{}&0.843511598585&{}:{}&0.214786146477&, \end{alignedat} \]
1a (013)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1a}}&{}\approx{}&0.077387626438&{}:{}&1.140083885251&{}:{}&-0.217471511689&,\\ A^*_{\mathbf{1a}}&{}\approx{}&0.000000000000&{}:{}&1.235712762933&{}:{}&-0.235712762933&,\\B^*_{\mathbf{1a}}&{}\approx{}&-0.552437750421&{}:{}&0.000000000000&{}:{}&1.552437750421&,\\C^*_{\mathbf{1a}}&{}\approx{}&0.063564219528&{}:{}&0.936435780472&{}:{}&0.000000000000&. \end{alignedat} \]
1a (013)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1a}}&{}\approx{}&-0.156518182688&{}:{}&0.768614941193&{}:{}&0.387903241495&,\\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.101723106389&{}:{}&0.932140100646&{}:{}&0.169583005743&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.234523192506&{}:{}&0.617180607251&{}:{}&0.617342585256&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.058297745062&{}:{}&0.843511598585&{}:{}&0.214786146477&,\\ A^*_{\mathbf{1a}}&{}\approx{}&0.000000000000&{}:{}&1.235712762933&{}:{}&-0.235712762933&,\\B^*_{\mathbf{1a}}&{}\approx{}&-0.552437750421&{}:{}&0.000000000000&{}:{}&1.552437750421&,\\C^*_{\mathbf{1a}}&{}\approx{}&0.063564219528&{}:{}&0.936435780472&{}:{}&0.000000000000&. \end{alignedat} \]
1a (013)