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 > 1b (103)

1b(103)

Malfatti circles

1b (103)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&1.000000000000&{}:{}&-1.000000000000&{}:{}&1.000000000000&, \\ P_{\mathbf{1b}}&{}\approx{}&0.962172809687&{}:{}&-0.069080899652&{}:{}&0.106908089965&, \\ P^-_{\mathbf{1b}}&{}\approx{}&0.958616393184&{}:{}&0.018441753337&{}:{}&0.022941853479&, \\ P^+_{\mathbf{1b}}&{}\approx{}&0.965166338260&{}:{}&-0.142751000530&{}:{}&0.177584662270&, \\ Q_{\mathbf{1b}}&{}\approx{}&1.217482258674&{}:{}&0.188345160884&{}:{}&-0.405827419558&, \\ I^\prime_{\mathbf{1b}}&{}\approx{}&0.938628576875&{}:{}&-0.251504769166&{}:{}&0.312876192292&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{1b}}\)
\(P^-_{\mathbf{1b}}\)
\(P^+_{\mathbf{1b}}\)
\(Q_{\mathbf{1b}}\)
\(I^\prime_{\mathbf{1b}}\)
1b (103)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1b}}&{}\approx{}&1.000000000000&{}:{}&-0.098076211353&{}:{}&0.098076211353&,\\B^\prime_{\mathbf{1b}}&{}\approx{}&0.455341801261&{}:{}&0.089316397477&{}:{}&0.455341801261&,\\C^\prime_{\mathbf{1b}}&{}\approx{}&4.098076211353&{}:{}&-4.098076211353&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.819615242271&{}:{}&-0.329422863406&{}:{}&0.509807621135&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.072410996238&{}:{}&-0.191567773597&{}:{}&0.119156777360&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.903476194491&{}:{}&-0.064866672287&{}:{}&0.161390477796&, \\ A^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.694362780359&{}:{}&0.136200941605&{}:{}&0.169436278036&,\\B^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.080081515006&{}:{}&-0.105930302945&{}:{}&0.025848787939&,\\C^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.895003474807&{}:{}&0.017217975236&{}:{}&0.087778549957&, \\ A^*_{\mathbf{1b}}&{}\approx{}&0.000000000000&{}:{}&-0.866025403784&{}:{}&1.866025403784&,\\B^*_{\mathbf{1b}}&{}\approx{}&1.500000000000&{}:{}&0.000000000000&{}:{}&-0.500000000000&,\\C^*_{\mathbf{1b}}&{}\approx{}&0.866025403784&{}:{}&0.133974596216&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1b}}}{B^\prime_{\mathbf{1b}}}{C^\prime_{\mathbf{1b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1b}}}{B^{\prime\prime}_{\mathbf{1b}}}{C^{\prime\prime}_{\mathbf{1b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1b}}}{B^{\prime\prime\prime}_{\mathbf{1b}}}{C^{\prime\prime\prime}_{\mathbf{1b}}}\)
\(\triangle{A^*_{\mathbf{1b}}}{B^*_{\mathbf{1b}}}{C^*_{\mathbf{1b}}}\)
1b (103)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1b}}}}&{}\approx{}&0.098076211353&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1b}}}}&{}\approx{}&0.455341801261&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1b}}}}&{}\approx{}&4.098076211353&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&1.000000000000&{}:{}&-1.000000000000&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{1b}}&{}\approx{}&1.000000000000&{}:{}&-0.098076211353&{}:{}&0.098076211353&,\\B^\prime_{\mathbf{1b}}&{}\approx{}&0.455341801261&{}:{}&0.089316397477&{}:{}&0.455341801261&,\\C^\prime_{\mathbf{1b}}&{}\approx{}&4.098076211353&{}:{}&-4.098076211353&{}:{}&1.000000000000&. \end{alignedat} \]
1b (103)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1b}}&{}\approx{}&0.962172809687&{}:{}&-0.069080899652&{}:{}&0.106908089965&,\\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.819615242271&{}:{}&-0.329422863406&{}:{}&0.509807621135&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.072410996238&{}:{}&-0.191567773597&{}:{}&0.119156777360&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.903476194491&{}:{}&-0.064866672287&{}:{}&0.161390477796&. \end{alignedat} \]
1b (103)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1b}}&{}\approx{}&0.958616393184&{}:{}&0.018441753337&{}:{}&0.022941853479&,\\ A^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.694362780359&{}:{}&0.136200941605&{}:{}&0.169436278036&,\\B^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.080081515006&{}:{}&-0.105930302945&{}:{}&0.025848787939&,\\C^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.895003474807&{}:{}&0.017217975236&{}:{}&0.087778549957&. \end{alignedat} \]
1b (103)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1b}}&{}\approx{}&0.965166338260&{}:{}&-0.142751000530&{}:{}&0.177584662270&,\\ A^\prime_{\mathbf{1b}}&{}\approx{}&1.000000000000&{}:{}&-0.098076211353&{}:{}&0.098076211353&,\\B^\prime_{\mathbf{1b}}&{}\approx{}&0.455341801261&{}:{}&0.089316397477&{}:{}&0.455341801261&,\\C^\prime_{\mathbf{1b}}&{}\approx{}&4.098076211353&{}:{}&-4.098076211353&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.819615242271&{}:{}&-0.329422863406&{}:{}&0.509807621135&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.072410996238&{}:{}&-0.191567773597&{}:{}&0.119156777360&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.903476194491&{}:{}&-0.064866672287&{}:{}&0.161390477796&, \end{alignedat} \]
1b (103)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1b}}&{}\approx{}&1.217482258674&{}:{}&0.188345160884&{}:{}&-0.405827419558&,\\ A^*_{\mathbf{1b}}&{}\approx{}&0.000000000000&{}:{}&-0.866025403784&{}:{}&1.866025403784&,\\B^*_{\mathbf{1b}}&{}\approx{}&1.500000000000&{}:{}&0.000000000000&{}:{}&-0.500000000000&,\\C^*_{\mathbf{1b}}&{}\approx{}&0.866025403784&{}:{}&0.133974596216&{}:{}&0.000000000000&. \end{alignedat} \]
1b (103)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1b}}&{}\approx{}&0.938628576875&{}:{}&-0.251504769166&{}:{}&0.312876192292&,\\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.819615242271&{}:{}&-0.329422863406&{}:{}&0.509807621135&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.072410996238&{}:{}&-0.191567773597&{}:{}&0.119156777360&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.903476194491&{}:{}&-0.064866672287&{}:{}&0.161390477796&,\\ A^*_{\mathbf{1b}}&{}\approx{}&0.000000000000&{}:{}&-0.866025403784&{}:{}&1.866025403784&,\\B^*_{\mathbf{1b}}&{}\approx{}&1.500000000000&{}:{}&0.000000000000&{}:{}&-0.500000000000&,\\C^*_{\mathbf{1b}}&{}\approx{}&0.866025403784&{}:{}&0.133974596216&{}:{}&0.000000000000&. \end{alignedat} \]
1b (103)