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 > 0c (110)

0c(110)

Malfatti circles

0c (110)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.732050807569&{}:{}&3.732050807569&{}:{}&-6.464101615138&, \\ P_{\mathbf{c}}&{}\approx{}&1.281796882634&{}:{}&1.281796882634&{}:{}&-1.563593765269&, \\ P^-_{\mathbf{c}}&{}\approx{}&0.965011002811&{}:{}&0.965011002811&{}:{}&-0.930022005622&, \\ P^+_{\mathbf{c}}&{}\approx{}&1.533499126792&{}:{}&1.533499126792&{}:{}&-2.066998253585&, \\ Q_{\mathbf{c}}&{}\approx{}&0.801550210820&{}:{}&0.801550210820&{}:{}&-0.603100421640&, \\ I^\prime_{\mathbf{c}}&{}\approx{}&1.830023832870&{}:{}&1.830023832870&{}:{}&-2.660047665740&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{c}}\)
\(P^-_{\mathbf{c}}\)
\(P^+_{\mathbf{c}}\)
\(Q_{\mathbf{c}}\)
\(I^\prime_{\mathbf{c}}\)
0c (110)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{c}}&{}\approx{}&1.577350269190&{}:{}&0.788675134595&{}:{}&-1.366025403784&,\\B^\prime_{\mathbf{c}}&{}\approx{}&0.788675134595&{}:{}&1.577350269190&{}:{}&-1.366025403784&,\\C^\prime_{\mathbf{c}}&{}\approx{}&13.790206641256&{}:{}&13.790206641256&{}:{}&-26.580413282512&, \\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.492020203758&{}:{}&2.238030305637&{}:{}&-2.730050509394&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&2.238030305637&{}:{}&1.492020203758&{}:{}&-2.730050509394&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.183012701892&{}:{}&1.183012701892&{}:{}&-1.366025403784&, \\ A^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.932387692461&{}:{}&1.864775384921&{}:{}&-1.797163077382&,\\B^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&1.864775384921&{}:{}&0.932387692461&{}:{}&-1.797163077382&,\\C^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.881853970395&{}:{}&0.881853970395&{}:{}&-0.763707940790&, \\ A^*_{\mathbf{c}}&{}\approx{}&0.000000000000&{}:{}&4.039058011260&{}:{}&-3.039058011260&,\\B^*_{\mathbf{c}}&{}\approx{}&4.039058011260&{}:{}&0.000000000000&{}:{}&-3.039058011260&,\\C^*_{\mathbf{c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{c}}}{B^\prime_{\mathbf{c}}}{C^\prime_{\mathbf{c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{c}}}{B^{\prime\prime}_{\mathbf{c}}}{C^{\prime\prime}_{\mathbf{c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{c}}}{B^{\prime\prime\prime}_{\mathbf{c}}}{C^{\prime\prime\prime}_{\mathbf{c}}}\)
\(\triangle{A^*_{\mathbf{c}}}{B^*_{\mathbf{c}}}{C^*_{\mathbf{c}}}\)
0c (110)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{c}}}}&{}\approx{}&0.211324865405&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{c}}}}&{}\approx{}&0.211324865405&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{c}}}}&{}\approx{}&3.695074732983&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.732050807569&{}:{}&3.732050807569&{}:{}&-6.464101615138&,\\ A^\prime_{\mathbf{c}}&{}\approx{}&1.577350269190&{}:{}&0.788675134595&{}:{}&-1.366025403784&,\\B^\prime_{\mathbf{c}}&{}\approx{}&0.788675134595&{}:{}&1.577350269190&{}:{}&-1.366025403784&,\\C^\prime_{\mathbf{c}}&{}\approx{}&13.790206641256&{}:{}&13.790206641256&{}:{}&-26.580413282512&. \end{alignedat} \]
0c (110)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{c}}&{}\approx{}&1.281796882634&{}:{}&1.281796882634&{}:{}&-1.563593765269&,\\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.492020203758&{}:{}&2.238030305637&{}:{}&-2.730050509394&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&2.238030305637&{}:{}&1.492020203758&{}:{}&-2.730050509394&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.183012701892&{}:{}&1.183012701892&{}:{}&-1.366025403784&. \end{alignedat} \]
0c (110)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{c}}&{}\approx{}&0.965011002811&{}:{}&0.965011002811&{}:{}&-0.930022005622&,\\ A^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.932387692461&{}:{}&1.864775384921&{}:{}&-1.797163077382&,\\B^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&1.864775384921&{}:{}&0.932387692461&{}:{}&-1.797163077382&,\\C^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.881853970395&{}:{}&0.881853970395&{}:{}&-0.763707940790&. \end{alignedat} \]
0c (110)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{c}}&{}\approx{}&1.533499126792&{}:{}&1.533499126792&{}:{}&-2.066998253585&,\\ A^\prime_{\mathbf{c}}&{}\approx{}&1.577350269190&{}:{}&0.788675134595&{}:{}&-1.366025403784&,\\B^\prime_{\mathbf{c}}&{}\approx{}&0.788675134595&{}:{}&1.577350269190&{}:{}&-1.366025403784&,\\C^\prime_{\mathbf{c}}&{}\approx{}&13.790206641256&{}:{}&13.790206641256&{}:{}&-26.580413282512&,\\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.492020203758&{}:{}&2.238030305637&{}:{}&-2.730050509394&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&2.238030305637&{}:{}&1.492020203758&{}:{}&-2.730050509394&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.183012701892&{}:{}&1.183012701892&{}:{}&-1.366025403784&, \end{alignedat} \]
0c (110)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{c}}&{}\approx{}&0.801550210820&{}:{}&0.801550210820&{}:{}&-0.603100421640&,\\ A^*_{\mathbf{c}}&{}\approx{}&0.000000000000&{}:{}&4.039058011260&{}:{}&-3.039058011260&,\\B^*_{\mathbf{c}}&{}\approx{}&4.039058011260&{}:{}&0.000000000000&{}:{}&-3.039058011260&,\\C^*_{\mathbf{c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
0c (110)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{c}}&{}\approx{}&1.830023832870&{}:{}&1.830023832870&{}:{}&-2.660047665740&,\\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.492020203758&{}:{}&2.238030305637&{}:{}&-2.730050509394&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&2.238030305637&{}:{}&1.492020203758&{}:{}&-2.730050509394&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.183012701892&{}:{}&1.183012701892&{}:{}&-1.366025403784&,\\ A^*_{\mathbf{c}}&{}\approx{}&0.000000000000&{}:{}&4.039058011260&{}:{}&-3.039058011260&,\\B^*_{\mathbf{c}}&{}\approx{}&4.039058011260&{}:{}&0.000000000000&{}:{}&-3.039058011260&,\\C^*_{\mathbf{c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
0c (110)