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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2c (130)

Malfatti circles

2c (130)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.000000000000&{}:{}&1.000000000000&{}:{}&-1.000000000000&, \\ P_{\mathbf{2c}}&{}\approx{}&0.962172809687&{}:{}&0.106908089965&{}:{}&-0.069080899652&, \\ P^-_{\mathbf{2c}}&{}\approx{}&0.958616393184&{}:{}&0.022941853479&{}:{}&0.018441753337&, \\ P^+_{\mathbf{2c}}&{}\approx{}&0.965166338260&{}:{}&0.177584662270&{}:{}&-0.142751000530&, \\ Q_{\mathbf{2c}}&{}\approx{}&1.217482258674&{}:{}&-0.405827419558&{}:{}&0.188345160884&, \\ I^\prime_{\mathbf{2c}}&{}\approx{}&0.938628576875&{}:{}&0.312876192292&{}:{}&-0.251504769166&, \end{alignedat} \]
\(I_{\mathbf{c}}\) Incenter
\(P_{\mathbf{2c}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{2c}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{2c}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{2c}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{2c}}\) Radical center of the Malfatti circles
2c (130)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{2c}}&{}\approx{}&1.000000000000&{}:{}&0.098076211353&{}:{}&-0.098076211353&,\\B^\prime_{\mathbf{2c}}&{}\approx{}&4.098076211353&{}:{}&1.000000000000&{}:{}&-4.098076211353&,\\C^\prime_{\mathbf{2c}}&{}\approx{}&0.455341801261&{}:{}&0.455341801261&{}:{}&0.089316397477&, \\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.819615242271&{}:{}&0.509807621135&{}:{}&-0.329422863406&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.903476194491&{}:{}&0.161390477796&{}:{}&-0.064866672287&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.072410996238&{}:{}&0.119156777360&{}:{}&-0.191567773597&, \\ A^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.694362780359&{}:{}&0.169436278036&{}:{}&0.136200941605&,\\B^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.895003474807&{}:{}&0.087778549957&{}:{}&0.017217975236&,\\C^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.080081515006&{}:{}&0.025848787939&{}:{}&-0.105930302945&, \\ A^*_{\mathbf{2c}}&{}\approx{}&0.000000000000&{}:{}&1.866025403784&{}:{}&-0.866025403784&,\\B^*_{\mathbf{2c}}&{}\approx{}&0.866025403784&{}:{}&0.000000000000&{}:{}&0.133974596216&,\\C^*_{\mathbf{2c}}&{}\approx{}&1.500000000000&{}:{}&-0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{2c}}}{B^\prime_{\mathbf{2c}}}{C^\prime_{\mathbf{2c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{2c}}}{B^{\prime\prime}_{\mathbf{2c}}}{C^{\prime\prime}_{\mathbf{2c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{2c}}}{B^{\prime\prime\prime}_{\mathbf{2c}}}{C^{\prime\prime\prime}_{\mathbf{2c}}}\)
\(\triangle{A^*_{\mathbf{2c}}}{B^*_{\mathbf{2c}}}{C^*_{\mathbf{2c}}}\)
2c (130)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{2c}}}}&{}\approx{}&0.098076211353&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{2c}}}}&{}\approx{}&4.098076211353&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{2c}}}}&{}\approx{}&0.455341801261&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.000000000000&{}:{}&1.000000000000&{}:{}&-1.000000000000&,\\ A^\prime_{\mathbf{2c}}&{}\approx{}&1.000000000000&{}:{}&0.098076211353&{}:{}&-0.098076211353&,\\B^\prime_{\mathbf{2c}}&{}\approx{}&4.098076211353&{}:{}&1.000000000000&{}:{}&-4.098076211353&,\\C^\prime_{\mathbf{2c}}&{}\approx{}&0.455341801261&{}:{}&0.455341801261&{}:{}&0.089316397477&. \end{alignedat} \]
2c (130)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{2c}}&{}\approx{}&0.962172809687&{}:{}&0.106908089965&{}:{}&-0.069080899652&,\\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.819615242271&{}:{}&0.509807621135&{}:{}&-0.329422863406&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.903476194491&{}:{}&0.161390477796&{}:{}&-0.064866672287&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.072410996238&{}:{}&0.119156777360&{}:{}&-0.191567773597&. \end{alignedat} \]
2c (130)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{2c}}&{}\approx{}&0.958616393184&{}:{}&0.022941853479&{}:{}&0.018441753337&,\\ A^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.694362780359&{}:{}&0.169436278036&{}:{}&0.136200941605&,\\B^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.895003474807&{}:{}&0.087778549957&{}:{}&0.017217975236&,\\C^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.080081515006&{}:{}&0.025848787939&{}:{}&-0.105930302945&. \end{alignedat} \]
2c (130)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{2c}}&{}\approx{}&0.965166338260&{}:{}&0.177584662270&{}:{}&-0.142751000530&,\\ A^\prime_{\mathbf{2c}}&{}\approx{}&1.000000000000&{}:{}&0.098076211353&{}:{}&-0.098076211353&,\\B^\prime_{\mathbf{2c}}&{}\approx{}&4.098076211353&{}:{}&1.000000000000&{}:{}&-4.098076211353&,\\C^\prime_{\mathbf{2c}}&{}\approx{}&0.455341801261&{}:{}&0.455341801261&{}:{}&0.089316397477&,\\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.819615242271&{}:{}&0.509807621135&{}:{}&-0.329422863406&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.903476194491&{}:{}&0.161390477796&{}:{}&-0.064866672287&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.072410996238&{}:{}&0.119156777360&{}:{}&-0.191567773597&, \end{alignedat} \]
2c (130)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{2c}}&{}\approx{}&1.217482258674&{}:{}&-0.405827419558&{}:{}&0.188345160884&,\\ A^*_{\mathbf{2c}}&{}\approx{}&0.000000000000&{}:{}&1.866025403784&{}:{}&-0.866025403784&,\\B^*_{\mathbf{2c}}&{}\approx{}&0.866025403784&{}:{}&0.000000000000&{}:{}&0.133974596216&,\\C^*_{\mathbf{2c}}&{}\approx{}&1.500000000000&{}:{}&-0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
2c (130)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{2c}}&{}\approx{}&0.938628576875&{}:{}&0.312876192292&{}:{}&-0.251504769166&,\\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.819615242271&{}:{}&0.509807621135&{}:{}&-0.329422863406&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.903476194491&{}:{}&0.161390477796&{}:{}&-0.064866672287&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.072410996238&{}:{}&0.119156777360&{}:{}&-0.191567773597&,\\ A^*_{\mathbf{2c}}&{}\approx{}&0.000000000000&{}:{}&1.866025403784&{}:{}&-0.866025403784&,\\B^*_{\mathbf{2c}}&{}\approx{}&0.866025403784&{}:{}&0.000000000000&{}:{}&0.133974596216&,\\C^*_{\mathbf{2c}}&{}\approx{}&1.500000000000&{}:{}&-0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
2c (130)