Derousseau's Generalization of the Malfatti circles

Isosceles Triangle with 120° Top Angle

\(A=30\degree\), \(B=30\degree\), \(C=120\degree\).


[Top] > Isosceles Triangle with 120° Top Angle > 3c (132)

3c(132)

Malfatti circles

3c (132)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.732050807569&{}:{}&3.732050807569&{}:{}&-6.464101615138&, \\ P_{\mathbf{3c}}&{}\approx{}&-0.140963072472&{}:{}&-0.406615064225&{}:{}&1.547578136697&, \\ P^-_{\mathbf{3c}}&{}\approx{}&0.092513363428&{}:{}&-0.157124360846&{}:{}&1.064610997418&, \\ P^+_{\mathbf{3c}}&{}\approx{}&-0.406447887546&{}:{}&-0.690309617784&{}:{}&2.096757505330&, \\ Q_{\mathbf{3c}}&{}\approx{}&0.641495994501&{}:{}&-1.089514469981&{}:{}&1.448018475480&, \\ I^\prime_{\mathbf{3c}}&{}\approx{}&-0.601583108703&{}:{}&-1.021726569529&{}:{}&2.623309678232&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{3c}}\)
\(P^-_{\mathbf{3c}}\)
\(P^+_{\mathbf{3c}}\)
\(Q_{\mathbf{3c}}\)
\(I^\prime_{\mathbf{3c}}\)
3c (132)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3c}}&{}\approx{}&0.359681312352&{}:{}&-0.874691593845&{}:{}&1.515010281493&,\\B^\prime_{\mathbf{3c}}&{}\approx{}&-29.713765674275&{}:{}&-20.751986157765&{}:{}&51.465751832040&,\\C^\prime_{\mathbf{3c}}&{}\approx{}&-0.366025403784&{}:{}&-0.366025403784&{}:{}&1.732050807569&, \\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.723142860312&{}:{}&-0.614091605346&{}:{}&2.337234465658&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.151933798438&{}:{}&-0.516087624230&{}:{}&1.668021422668&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.500283315696&{}:{}&-1.443092357278&{}:{}&2.943375672974&, \\ A^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.329635435207&{}:{}&-0.230216192169&{}:{}&1.559851627376&,\\B^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.099247762266&{}:{}&-0.241355834681&{}:{}&1.142108072415&,\\C^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.284600744942&{}:{}&-0.483364872796&{}:{}&1.198764127855&, \\ A^*_{\mathbf{3c}}&{}\approx{}&0.000000000000&{}:{}&-3.039058011260&{}:{}&4.039058011260&,\\B^*_{\mathbf{3c}}&{}\approx{}&0.307007203691&{}:{}&0.000000000000&{}:{}&0.692992796309&,\\C^*_{\mathbf{3c}}&{}\approx{}&-1.431851652578&{}:{}&2.431851652578&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3c}}}{B^\prime_{\mathbf{3c}}}{C^\prime_{\mathbf{3c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3c}}}{B^{\prime\prime}_{\mathbf{3c}}}{C^{\prime\prime}_{\mathbf{3c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3c}}}{B^{\prime\prime\prime}_{\mathbf{3c}}}{C^{\prime\prime\prime}_{\mathbf{3c}}}\)
\(\triangle{A^*_{\mathbf{3c}}}{B^*_{\mathbf{3c}}}{C^*_{\mathbf{3c}}}\)
3c (132)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3c}}}}&{}\approx{}&-0.234372906197&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3c}}}}&{}\approx{}&-7.961779516510&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3c}}}}&{}\approx{}&-0.098076211353&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.732050807569&{}:{}&3.732050807569&{}:{}&-6.464101615138&,\\ A^\prime_{\mathbf{3c}}&{}\approx{}&0.359681312352&{}:{}&-0.874691593845&{}:{}&1.515010281493&,\\B^\prime_{\mathbf{3c}}&{}\approx{}&-29.713765674275&{}:{}&-20.751986157765&{}:{}&51.465751832040&,\\C^\prime_{\mathbf{3c}}&{}\approx{}&-0.366025403784&{}:{}&-0.366025403784&{}:{}&1.732050807569&. \end{alignedat} \]
3c (132)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3c}}&{}\approx{}&-0.140963072472&{}:{}&-0.406615064225&{}:{}&1.547578136697&,\\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.723142860312&{}:{}&-0.614091605346&{}:{}&2.337234465658&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.151933798438&{}:{}&-0.516087624230&{}:{}&1.668021422668&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.500283315696&{}:{}&-1.443092357278&{}:{}&2.943375672974&. \end{alignedat} \]
3c (132)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3c}}&{}\approx{}&0.092513363428&{}:{}&-0.157124360846&{}:{}&1.064610997418&,\\ A^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.329635435207&{}:{}&-0.230216192169&{}:{}&1.559851627376&,\\B^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.099247762266&{}:{}&-0.241355834681&{}:{}&1.142108072415&,\\C^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.284600744942&{}:{}&-0.483364872796&{}:{}&1.198764127855&. \end{alignedat} \]
3c (132)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3c}}&{}\approx{}&-0.406447887546&{}:{}&-0.690309617784&{}:{}&2.096757505330&,\\ A^\prime_{\mathbf{3c}}&{}\approx{}&0.359681312352&{}:{}&-0.874691593845&{}:{}&1.515010281493&,\\B^\prime_{\mathbf{3c}}&{}\approx{}&-29.713765674275&{}:{}&-20.751986157765&{}:{}&51.465751832040&,\\C^\prime_{\mathbf{3c}}&{}\approx{}&-0.366025403784&{}:{}&-0.366025403784&{}:{}&1.732050807569&,\\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.723142860312&{}:{}&-0.614091605346&{}:{}&2.337234465658&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.151933798438&{}:{}&-0.516087624230&{}:{}&1.668021422668&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.500283315696&{}:{}&-1.443092357278&{}:{}&2.943375672974&, \end{alignedat} \]
3c (132)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3c}}&{}\approx{}&0.641495994501&{}:{}&-1.089514469981&{}:{}&1.448018475480&,\\ A^*_{\mathbf{3c}}&{}\approx{}&0.000000000000&{}:{}&-3.039058011260&{}:{}&4.039058011260&,\\B^*_{\mathbf{3c}}&{}\approx{}&0.307007203691&{}:{}&0.000000000000&{}:{}&0.692992796309&,\\C^*_{\mathbf{3c}}&{}\approx{}&-1.431851652578&{}:{}&2.431851652578&{}:{}&0.000000000000&. \end{alignedat} \]
3c (132)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3c}}&{}\approx{}&-0.601583108703&{}:{}&-1.021726569529&{}:{}&2.623309678232&,\\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.723142860312&{}:{}&-0.614091605346&{}:{}&2.337234465658&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.151933798438&{}:{}&-0.516087624230&{}:{}&1.668021422668&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.500283315696&{}:{}&-1.443092357278&{}:{}&2.943375672974&,\\ A^*_{\mathbf{3c}}&{}\approx{}&0.000000000000&{}:{}&-3.039058011260&{}:{}&4.039058011260&,\\B^*_{\mathbf{3c}}&{}\approx{}&0.307007203691&{}:{}&0.000000000000&{}:{}&0.692992796309&,\\C^*_{\mathbf{3c}}&{}\approx{}&-1.431851652578&{}:{}&2.431851652578&{}:{}&0.000000000000&. \end{alignedat} \]
3c (132)