Derousseau's Generalization of the Malfatti circles

Equilateral Triangle

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


[Top] > Equilateral Triangle > 4b (301)

4b(301)

Malfatti circles

4b (301)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&1.000000000000&{}:{}&-1.000000000000&{}:{}&1.000000000000&, \\ P_{\mathbf{4b}}&{}\approx{}&0.106908089965&{}:{}&-0.069080899652&{}:{}&0.962172809687&, \\ P^-_{\mathbf{4b}}&{}\approx{}&0.022941853479&{}:{}&0.018441753337&{}:{}&0.958616393184&, \\ P^+_{\mathbf{4b}}&{}\approx{}&0.177584662270&{}:{}&-0.142751000530&{}:{}&0.965166338260&, \\ Q_{\mathbf{4b}}&{}\approx{}&-0.405827419558&{}:{}&0.188345160884&{}:{}&1.217482258674&, \\ I^\prime_{\mathbf{4b}}&{}\approx{}&0.312876192292&{}:{}&-0.251504769166&{}:{}&0.938628576875&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{4b}}\)
\(P^-_{\mathbf{4b}}\)
\(P^+_{\mathbf{4b}}\)
\(Q_{\mathbf{4b}}\)
\(I^\prime_{\mathbf{4b}}\)
4b (301)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4b}}&{}\approx{}&1.000000000000&{}:{}&-4.098076211353&{}:{}&4.098076211353&,\\B^\prime_{\mathbf{4b}}&{}\approx{}&0.455341801261&{}:{}&0.089316397477&{}:{}&0.455341801261&,\\C^\prime_{\mathbf{4b}}&{}\approx{}&0.098076211353&{}:{}&-0.098076211353&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.161390477796&{}:{}&-0.064866672287&{}:{}&0.903476194491&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.119156777360&{}:{}&-0.191567773597&{}:{}&1.072410996238&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.509807621135&{}:{}&-0.329422863406&{}:{}&0.819615242271&, \\ A^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.087778549957&{}:{}&0.017217975236&{}:{}&0.895003474807&,\\B^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.025848787939&{}:{}&-0.105930302945&{}:{}&1.080081515006&,\\C^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.169436278036&{}:{}&0.136200941605&{}:{}&0.694362780359&, \\ A^*_{\mathbf{4b}}&{}\approx{}&0.000000000000&{}:{}&0.133974596216&{}:{}&0.866025403784&,\\B^*_{\mathbf{4b}}&{}\approx{}&-0.500000000000&{}:{}&0.000000000000&{}:{}&1.500000000000&,\\C^*_{\mathbf{4b}}&{}\approx{}&1.866025403784&{}:{}&-0.866025403784&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4b}}}{B^\prime_{\mathbf{4b}}}{C^\prime_{\mathbf{4b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4b}}}{B^{\prime\prime}_{\mathbf{4b}}}{C^{\prime\prime}_{\mathbf{4b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4b}}}{B^{\prime\prime\prime}_{\mathbf{4b}}}{C^{\prime\prime\prime}_{\mathbf{4b}}}\)
\(\triangle{A^*_{\mathbf{4b}}}{B^*_{\mathbf{4b}}}{C^*_{\mathbf{4b}}}\)
4b (301)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4b}}}}&{}\approx{}&4.098076211353&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4b}}}}&{}\approx{}&0.455341801261&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4b}}}}&{}\approx{}&0.098076211353&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&1.000000000000&{}:{}&-1.000000000000&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{4b}}&{}\approx{}&1.000000000000&{}:{}&-4.098076211353&{}:{}&4.098076211353&,\\B^\prime_{\mathbf{4b}}&{}\approx{}&0.455341801261&{}:{}&0.089316397477&{}:{}&0.455341801261&,\\C^\prime_{\mathbf{4b}}&{}\approx{}&0.098076211353&{}:{}&-0.098076211353&{}:{}&1.000000000000&. \end{alignedat} \]
4b (301)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4b}}&{}\approx{}&0.106908089965&{}:{}&-0.069080899652&{}:{}&0.962172809687&,\\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.161390477796&{}:{}&-0.064866672287&{}:{}&0.903476194491&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.119156777360&{}:{}&-0.191567773597&{}:{}&1.072410996238&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.509807621135&{}:{}&-0.329422863406&{}:{}&0.819615242271&. \end{alignedat} \]
4b (301)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4b}}&{}\approx{}&0.022941853479&{}:{}&0.018441753337&{}:{}&0.958616393184&,\\ A^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.087778549957&{}:{}&0.017217975236&{}:{}&0.895003474807&,\\B^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.025848787939&{}:{}&-0.105930302945&{}:{}&1.080081515006&,\\C^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.169436278036&{}:{}&0.136200941605&{}:{}&0.694362780359&. \end{alignedat} \]
4b (301)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4b}}&{}\approx{}&0.177584662270&{}:{}&-0.142751000530&{}:{}&0.965166338260&,\\ A^\prime_{\mathbf{4b}}&{}\approx{}&1.000000000000&{}:{}&-4.098076211353&{}:{}&4.098076211353&,\\B^\prime_{\mathbf{4b}}&{}\approx{}&0.455341801261&{}:{}&0.089316397477&{}:{}&0.455341801261&,\\C^\prime_{\mathbf{4b}}&{}\approx{}&0.098076211353&{}:{}&-0.098076211353&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.161390477796&{}:{}&-0.064866672287&{}:{}&0.903476194491&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.119156777360&{}:{}&-0.191567773597&{}:{}&1.072410996238&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.509807621135&{}:{}&-0.329422863406&{}:{}&0.819615242271&, \end{alignedat} \]
4b (301)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4b}}&{}\approx{}&-0.405827419558&{}:{}&0.188345160884&{}:{}&1.217482258674&,\\ A^*_{\mathbf{4b}}&{}\approx{}&0.000000000000&{}:{}&0.133974596216&{}:{}&0.866025403784&,\\B^*_{\mathbf{4b}}&{}\approx{}&-0.500000000000&{}:{}&0.000000000000&{}:{}&1.500000000000&,\\C^*_{\mathbf{4b}}&{}\approx{}&1.866025403784&{}:{}&-0.866025403784&{}:{}&0.000000000000&. \end{alignedat} \]
4b (301)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4b}}&{}\approx{}&0.312876192292&{}:{}&-0.251504769166&{}:{}&0.938628576875&,\\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.161390477796&{}:{}&-0.064866672287&{}:{}&0.903476194491&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.119156777360&{}:{}&-0.191567773597&{}:{}&1.072410996238&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.509807621135&{}:{}&-0.329422863406&{}:{}&0.819615242271&,\\ A^*_{\mathbf{4b}}&{}\approx{}&0.000000000000&{}:{}&0.133974596216&{}:{}&0.866025403784&,\\B^*_{\mathbf{4b}}&{}\approx{}&-0.500000000000&{}:{}&0.000000000000&{}:{}&1.500000000000&,\\C^*_{\mathbf{4b}}&{}\approx{}&1.866025403784&{}:{}&-0.866025403784&{}:{}&0.000000000000&. \end{alignedat} \]
4b (301)