Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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5a(213)

Malfatti circles

5a (213)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.333333333333&{}:{}&0.555555555556&{}:{}&0.777777777778&, \\ P_{\mathbf{5a}}&{}\approx{}&1.044405331076&{}:{}&-0.000320809529&{}:{}&-0.044084521547&, \\ P^-_{\mathbf{5a}}&{}\approx{}&0.855657824819&{}:{}&0.075833166843&{}:{}&0.068509008337&, \\ P^+_{\mathbf{5a}}&{}\approx{}&1.304386723298&{}:{}&-0.105215529429&{}:{}&-0.199171193870&, \\ Q_{\mathbf{5a}}&{}\approx{}&1.404440927829&{}:{}&0.095331888542&{}:{}&-0.499772816371&, \\ I^\prime_{\mathbf{5a}}&{}\approx{}&1.507063777257&{}:{}&-0.034099271151&{}:{}&-0.472964506106&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{5a}}\)
\(P^-_{\mathbf{5a}}\)
\(P^+_{\mathbf{5a}}\)
\(Q_{\mathbf{5a}}\)
\(I^\prime_{\mathbf{5a}}\)
5a (213)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5a}}&{}\approx{}&1.271420142185&{}:{}&-0.113091725910&{}:{}&-0.158328416275&,\\B^\prime_{\mathbf{5a}}&{}\approx{}&3.055575084545&{}:{}&5.074100112726&{}:{}&-7.129675197271&,\\C^\prime_{\mathbf{5a}}&{}\approx{}&1.199685227256&{}:{}&-1.999475378759&{}:{}&1.799790151504&, \\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.722916118449&{}:{}&-0.005222759835&{}:{}&-0.717693358614&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.267579968152&{}:{}&-0.214075208309&{}:{}&-0.053504759844&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.310180060729&{}:{}&-0.000402447437&{}:{}&-0.309777613293&, \\ A^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&0.240337089970&{}:{}&0.399104725482&{}:{}&0.360558184548&,\\B^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.008962863918&{}:{}&-0.089746377200&{}:{}&0.080783513282&,\\C^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.037240263631&{}:{}&0.091926014917&{}:{}&-0.129166278548&, \\ A^*_{\mathbf{5a}}&{}\approx{}&0.000000000000&{}:{}&-0.235712762933&{}:{}&1.235712762933&,\\B^*_{\mathbf{5a}}&{}\approx{}&1.552437750421&{}:{}&0.000000000000&{}:{}&-0.552437750421&,\\C^*_{\mathbf{5a}}&{}\approx{}&0.936435780472&{}:{}&0.063564219528&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5a}}}{B^\prime_{\mathbf{5a}}}{C^\prime_{\mathbf{5a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5a}}}{B^{\prime\prime}_{\mathbf{5a}}}{C^{\prime\prime}_{\mathbf{5a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5a}}}{B^{\prime\prime\prime}_{\mathbf{5a}}}{C^{\prime\prime\prime}_{\mathbf{5a}}}\)
\(\triangle{A^*_{\mathbf{5a}}}{B^*_{\mathbf{5a}}}{C^*_{\mathbf{5a}}}\)
5a (213)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5a}}}}&{}\approx{}&-0.203565106639&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5a}}}}&{}\approx{}&-9.166725253634&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5a}}}}&{}\approx{}&-3.599055681767&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.333333333333&{}:{}&0.555555555556&{}:{}&0.777777777778&,\\ A^\prime_{\mathbf{5a}}&{}\approx{}&1.271420142185&{}:{}&-0.113091725910&{}:{}&-0.158328416275&,\\B^\prime_{\mathbf{5a}}&{}\approx{}&3.055575084545&{}:{}&5.074100112726&{}:{}&-7.129675197271&,\\C^\prime_{\mathbf{5a}}&{}\approx{}&1.199685227256&{}:{}&-1.999475378759&{}:{}&1.799790151504&. \end{alignedat} \]
5a (213)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5a}}&{}\approx{}&1.044405331076&{}:{}&-0.000320809529&{}:{}&-0.044084521547&,\\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.722916118449&{}:{}&-0.005222759835&{}:{}&-0.717693358614&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.267579968152&{}:{}&-0.214075208309&{}:{}&-0.053504759844&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.310180060729&{}:{}&-0.000402447437&{}:{}&-0.309777613293&. \end{alignedat} \]
5a (213)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5a}}&{}\approx{}&0.855657824819&{}:{}&0.075833166843&{}:{}&0.068509008337&,\\ A^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&0.240337089970&{}:{}&0.399104725482&{}:{}&0.360558184548&,\\B^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.008962863918&{}:{}&-0.089746377200&{}:{}&0.080783513282&,\\C^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.037240263631&{}:{}&0.091926014917&{}:{}&-0.129166278548&. \end{alignedat} \]
5a (213)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5a}}&{}\approx{}&1.304386723298&{}:{}&-0.105215529429&{}:{}&-0.199171193870&,\\ A^\prime_{\mathbf{5a}}&{}\approx{}&1.271420142185&{}:{}&-0.113091725910&{}:{}&-0.158328416275&,\\B^\prime_{\mathbf{5a}}&{}\approx{}&3.055575084545&{}:{}&5.074100112726&{}:{}&-7.129675197271&,\\C^\prime_{\mathbf{5a}}&{}\approx{}&1.199685227256&{}:{}&-1.999475378759&{}:{}&1.799790151504&,\\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.722916118449&{}:{}&-0.005222759835&{}:{}&-0.717693358614&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.267579968152&{}:{}&-0.214075208309&{}:{}&-0.053504759844&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.310180060729&{}:{}&-0.000402447437&{}:{}&-0.309777613293&, \end{alignedat} \]
5a (213)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5a}}&{}\approx{}&1.404440927829&{}:{}&0.095331888542&{}:{}&-0.499772816371&,\\ A^*_{\mathbf{5a}}&{}\approx{}&0.000000000000&{}:{}&-0.235712762933&{}:{}&1.235712762933&,\\B^*_{\mathbf{5a}}&{}\approx{}&1.552437750421&{}:{}&0.000000000000&{}:{}&-0.552437750421&,\\C^*_{\mathbf{5a}}&{}\approx{}&0.936435780472&{}:{}&0.063564219528&{}:{}&0.000000000000&. \end{alignedat} \]
5a (213)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5a}}&{}\approx{}&1.507063777257&{}:{}&-0.034099271151&{}:{}&-0.472964506106&,\\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.722916118449&{}:{}&-0.005222759835&{}:{}&-0.717693358614&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.267579968152&{}:{}&-0.214075208309&{}:{}&-0.053504759844&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.310180060729&{}:{}&-0.000402447437&{}:{}&-0.309777613293&,\\ A^*_{\mathbf{5a}}&{}\approx{}&0.000000000000&{}:{}&-0.235712762933&{}:{}&1.235712762933&,\\B^*_{\mathbf{5a}}&{}\approx{}&1.552437750421&{}:{}&0.000000000000&{}:{}&-0.552437750421&,\\C^*_{\mathbf{5a}}&{}\approx{}&0.936435780472&{}:{}&0.063564219528&{}:{}&0.000000000000&. \end{alignedat} \]
5a (213)