Derousseau's Generalization of the Malfatti circles

Equilateral Triangle

\(a:b:c=1:1:1\), \(A=60\degree\), \(B=60\degree\), \(C=60\degree\).


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1a(013)

Malfatti circles

1a (013)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-1.000000000000&{}:{}&1.000000000000&{}:{}&1.000000000000&, \\ P_{\mathbf{1a}}&{}\approx{}&-0.069080899652&{}:{}&0.962172809687&{}:{}&0.106908089965&, \\ P^-_{\mathbf{1a}}&{}\approx{}&0.018441753337&{}:{}&0.958616393184&{}:{}&0.022941853479&, \\ P^+_{\mathbf{1a}}&{}\approx{}&-0.142751000530&{}:{}&0.965166338260&{}:{}&0.177584662270&, \\ Q_{\mathbf{1a}}&{}\approx{}&0.188345160884&{}:{}&1.217482258674&{}:{}&-0.405827419558&, \\ I^\prime_{\mathbf{1a}}&{}\approx{}&-0.251504769166&{}:{}&0.938628576875&{}:{}&0.312876192292&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{1a}}\)
\(P^-_{\mathbf{1a}}\)
\(P^+_{\mathbf{1a}}\)
\(Q_{\mathbf{1a}}\)
\(I^\prime_{\mathbf{1a}}\)
1a (013)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1a}}&{}\approx{}&0.089316397477&{}:{}&0.455341801261&{}:{}&0.455341801261&,\\B^\prime_{\mathbf{1a}}&{}\approx{}&-0.098076211353&{}:{}&1.000000000000&{}:{}&0.098076211353&,\\C^\prime_{\mathbf{1a}}&{}\approx{}&-4.098076211353&{}:{}&4.098076211353&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.191567773597&{}:{}&1.072410996238&{}:{}&0.119156777360&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.329422863406&{}:{}&0.819615242271&{}:{}&0.509807621135&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.064866672287&{}:{}&0.903476194491&{}:{}&0.161390477796&, \\ A^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.105930302945&{}:{}&1.080081515006&{}:{}&0.025848787939&,\\B^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&0.136200941605&{}:{}&0.694362780359&{}:{}&0.169436278036&,\\C^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&0.017217975236&{}:{}&0.895003474807&{}:{}&0.087778549957&, \\ A^*_{\mathbf{1a}}&{}\approx{}&0.000000000000&{}:{}&1.500000000000&{}:{}&-0.500000000000&,\\B^*_{\mathbf{1a}}&{}\approx{}&-0.866025403784&{}:{}&0.000000000000&{}:{}&1.866025403784&,\\C^*_{\mathbf{1a}}&{}\approx{}&0.133974596216&{}:{}&0.866025403784&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1a}}}{B^\prime_{\mathbf{1a}}}{C^\prime_{\mathbf{1a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1a}}}{B^{\prime\prime}_{\mathbf{1a}}}{C^{\prime\prime}_{\mathbf{1a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1a}}}{B^{\prime\prime\prime}_{\mathbf{1a}}}{C^{\prime\prime\prime}_{\mathbf{1a}}}\)
\(\triangle{A^*_{\mathbf{1a}}}{B^*_{\mathbf{1a}}}{C^*_{\mathbf{1a}}}\)
1a (013)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1a}}}}&{}\approx{}&0.455341801261&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1a}}}}&{}\approx{}&0.098076211353&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1a}}}}&{}\approx{}&4.098076211353&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-1.000000000000&{}:{}&1.000000000000&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{1a}}&{}\approx{}&0.089316397477&{}:{}&0.455341801261&{}:{}&0.455341801261&,\\B^\prime_{\mathbf{1a}}&{}\approx{}&-0.098076211353&{}:{}&1.000000000000&{}:{}&0.098076211353&,\\C^\prime_{\mathbf{1a}}&{}\approx{}&-4.098076211353&{}:{}&4.098076211353&{}:{}&1.000000000000&. \end{alignedat} \]
1a (013)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1a}}&{}\approx{}&-0.069080899652&{}:{}&0.962172809687&{}:{}&0.106908089965&,\\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.191567773597&{}:{}&1.072410996238&{}:{}&0.119156777360&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.329422863406&{}:{}&0.819615242271&{}:{}&0.509807621135&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.064866672287&{}:{}&0.903476194491&{}:{}&0.161390477796&. \end{alignedat} \]
1a (013)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1a}}&{}\approx{}&0.018441753337&{}:{}&0.958616393184&{}:{}&0.022941853479&,\\ A^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.105930302945&{}:{}&1.080081515006&{}:{}&0.025848787939&,\\B^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&0.136200941605&{}:{}&0.694362780359&{}:{}&0.169436278036&,\\C^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&0.017217975236&{}:{}&0.895003474807&{}:{}&0.087778549957&. \end{alignedat} \]
1a (013)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1a}}&{}\approx{}&-0.142751000530&{}:{}&0.965166338260&{}:{}&0.177584662270&,\\ A^\prime_{\mathbf{1a}}&{}\approx{}&0.089316397477&{}:{}&0.455341801261&{}:{}&0.455341801261&,\\B^\prime_{\mathbf{1a}}&{}\approx{}&-0.098076211353&{}:{}&1.000000000000&{}:{}&0.098076211353&,\\C^\prime_{\mathbf{1a}}&{}\approx{}&-4.098076211353&{}:{}&4.098076211353&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.191567773597&{}:{}&1.072410996238&{}:{}&0.119156777360&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.329422863406&{}:{}&0.819615242271&{}:{}&0.509807621135&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.064866672287&{}:{}&0.903476194491&{}:{}&0.161390477796&, \end{alignedat} \]
1a (013)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1a}}&{}\approx{}&0.188345160884&{}:{}&1.217482258674&{}:{}&-0.405827419558&,\\ A^*_{\mathbf{1a}}&{}\approx{}&0.000000000000&{}:{}&1.500000000000&{}:{}&-0.500000000000&,\\B^*_{\mathbf{1a}}&{}\approx{}&-0.866025403784&{}:{}&0.000000000000&{}:{}&1.866025403784&,\\C^*_{\mathbf{1a}}&{}\approx{}&0.133974596216&{}:{}&0.866025403784&{}:{}&0.000000000000&. \end{alignedat} \]
1a (013)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1a}}&{}\approx{}&-0.251504769166&{}:{}&0.938628576875&{}:{}&0.312876192292&,\\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.191567773597&{}:{}&1.072410996238&{}:{}&0.119156777360&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.329422863406&{}:{}&0.819615242271&{}:{}&0.509807621135&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.064866672287&{}:{}&0.903476194491&{}:{}&0.161390477796&,\\ A^*_{\mathbf{1a}}&{}\approx{}&0.000000000000&{}:{}&1.500000000000&{}:{}&-0.500000000000&,\\B^*_{\mathbf{1a}}&{}\approx{}&-0.866025403784&{}:{}&0.000000000000&{}:{}&1.866025403784&,\\C^*_{\mathbf{1a}}&{}\approx{}&0.133974596216&{}:{}&0.866025403784&{}:{}&0.000000000000&. \end{alignedat} \]
1a (013)