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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4c(310)

Malfatti circles

4c (310)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.000000000000&{}:{}&1.000000000000&{}:{}&-1.000000000000&, \\ P_{\mathbf{4c}}&{}\approx{}&0.106908089965&{}:{}&0.962172809687&{}:{}&-0.069080899652&, \\ P^-_{\mathbf{4c}}&{}\approx{}&0.022941853479&{}:{}&0.958616393184&{}:{}&0.018441753337&, \\ P^+_{\mathbf{4c}}&{}\approx{}&0.177584662270&{}:{}&0.965166338260&{}:{}&-0.142751000530&, \\ Q_{\mathbf{4c}}&{}\approx{}&-0.405827419558&{}:{}&1.217482258674&{}:{}&0.188345160884&, \\ I^\prime_{\mathbf{4c}}&{}\approx{}&0.312876192292&{}:{}&0.938628576875&{}:{}&-0.251504769166&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{4c}}\)
\(P^-_{\mathbf{4c}}\)
\(P^+_{\mathbf{4c}}\)
\(Q_{\mathbf{4c}}\)
\(I^\prime_{\mathbf{4c}}\)
4c (310)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4c}}&{}\approx{}&1.000000000000&{}:{}&4.098076211353&{}:{}&-4.098076211353&,\\B^\prime_{\mathbf{4c}}&{}\approx{}&0.098076211353&{}:{}&1.000000000000&{}:{}&-0.098076211353&,\\C^\prime_{\mathbf{4c}}&{}\approx{}&0.455341801261&{}:{}&0.455341801261&{}:{}&0.089316397477&, \\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.161390477796&{}:{}&0.903476194491&{}:{}&-0.064866672287&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.509807621135&{}:{}&0.819615242271&{}:{}&-0.329422863406&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.119156777360&{}:{}&1.072410996238&{}:{}&-0.191567773597&, \\ A^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.087778549957&{}:{}&0.895003474807&{}:{}&0.017217975236&,\\B^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.169436278036&{}:{}&0.694362780359&{}:{}&0.136200941605&,\\C^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.025848787939&{}:{}&1.080081515006&{}:{}&-0.105930302945&, \\ A^*_{\mathbf{4c}}&{}\approx{}&0.000000000000&{}:{}&0.866025403784&{}:{}&0.133974596216&,\\B^*_{\mathbf{4c}}&{}\approx{}&1.866025403784&{}:{}&0.000000000000&{}:{}&-0.866025403784&,\\C^*_{\mathbf{4c}}&{}\approx{}&-0.500000000000&{}:{}&1.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4c}}}{B^\prime_{\mathbf{4c}}}{C^\prime_{\mathbf{4c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4c}}}{B^{\prime\prime}_{\mathbf{4c}}}{C^{\prime\prime}_{\mathbf{4c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4c}}}{B^{\prime\prime\prime}_{\mathbf{4c}}}{C^{\prime\prime\prime}_{\mathbf{4c}}}\)
\(\triangle{A^*_{\mathbf{4c}}}{B^*_{\mathbf{4c}}}{C^*_{\mathbf{4c}}}\)
4c (310)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4c}}}}&{}\approx{}&4.098076211353&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4c}}}}&{}\approx{}&0.098076211353&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4c}}}}&{}\approx{}&0.455341801261&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.000000000000&{}:{}&1.000000000000&{}:{}&-1.000000000000&,\\ A^\prime_{\mathbf{4c}}&{}\approx{}&1.000000000000&{}:{}&4.098076211353&{}:{}&-4.098076211353&,\\B^\prime_{\mathbf{4c}}&{}\approx{}&0.098076211353&{}:{}&1.000000000000&{}:{}&-0.098076211353&,\\C^\prime_{\mathbf{4c}}&{}\approx{}&0.455341801261&{}:{}&0.455341801261&{}:{}&0.089316397477&. \end{alignedat} \]
4c (310)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4c}}&{}\approx{}&0.106908089965&{}:{}&0.962172809687&{}:{}&-0.069080899652&,\\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.161390477796&{}:{}&0.903476194491&{}:{}&-0.064866672287&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.509807621135&{}:{}&0.819615242271&{}:{}&-0.329422863406&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.119156777360&{}:{}&1.072410996238&{}:{}&-0.191567773597&. \end{alignedat} \]
4c (310)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4c}}&{}\approx{}&0.022941853479&{}:{}&0.958616393184&{}:{}&0.018441753337&,\\ A^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.087778549957&{}:{}&0.895003474807&{}:{}&0.017217975236&,\\B^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.169436278036&{}:{}&0.694362780359&{}:{}&0.136200941605&,\\C^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.025848787939&{}:{}&1.080081515006&{}:{}&-0.105930302945&. \end{alignedat} \]
4c (310)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4c}}&{}\approx{}&0.177584662270&{}:{}&0.965166338260&{}:{}&-0.142751000530&,\\ A^\prime_{\mathbf{4c}}&{}\approx{}&1.000000000000&{}:{}&4.098076211353&{}:{}&-4.098076211353&,\\B^\prime_{\mathbf{4c}}&{}\approx{}&0.098076211353&{}:{}&1.000000000000&{}:{}&-0.098076211353&,\\C^\prime_{\mathbf{4c}}&{}\approx{}&0.455341801261&{}:{}&0.455341801261&{}:{}&0.089316397477&,\\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.161390477796&{}:{}&0.903476194491&{}:{}&-0.064866672287&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.509807621135&{}:{}&0.819615242271&{}:{}&-0.329422863406&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.119156777360&{}:{}&1.072410996238&{}:{}&-0.191567773597&, \end{alignedat} \]
4c (310)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4c}}&{}\approx{}&-0.405827419558&{}:{}&1.217482258674&{}:{}&0.188345160884&,\\ A^*_{\mathbf{4c}}&{}\approx{}&0.000000000000&{}:{}&0.866025403784&{}:{}&0.133974596216&,\\B^*_{\mathbf{4c}}&{}\approx{}&1.866025403784&{}:{}&0.000000000000&{}:{}&-0.866025403784&,\\C^*_{\mathbf{4c}}&{}\approx{}&-0.500000000000&{}:{}&1.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
4c (310)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4c}}&{}\approx{}&0.312876192292&{}:{}&0.938628576875&{}:{}&-0.251504769166&,\\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.161390477796&{}:{}&0.903476194491&{}:{}&-0.064866672287&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.509807621135&{}:{}&0.819615242271&{}:{}&-0.329422863406&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.119156777360&{}:{}&1.072410996238&{}:{}&-0.191567773597&,\\ A^*_{\mathbf{4c}}&{}\approx{}&0.000000000000&{}:{}&0.866025403784&{}:{}&0.133974596216&,\\B^*_{\mathbf{4c}}&{}\approx{}&1.866025403784&{}:{}&0.000000000000&{}:{}&-0.866025403784&,\\C^*_{\mathbf{4c}}&{}\approx{}&-0.500000000000&{}:{}&1.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
4c (310)