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 > 2c (130)

2c(130)

Malfatti circles

2c (130)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.732050807569&{}:{}&3.732050807569&{}:{}&-6.464101615138&, \\ P_{\mathbf{2c}}&{}\approx{}&1.082502636859&{}:{}&0.375276056093&{}:{}&-0.457778692951&, \\ P^-_{\mathbf{2c}}&{}\approx{}&0.922566064186&{}:{}&0.172648659120&{}:{}&-0.095214723306&, \\ P^+_{\mathbf{2c}}&{}\approx{}&1.225210460522&{}:{}&0.556075946659&{}:{}&-0.781286407181&, \\ Q_{\mathbf{2c}}&{}\approx{}&1.170650750933&{}:{}&-0.689268282684&{}:{}&0.518617531751&, \\ I^\prime_{\mathbf{2c}}&{}\approx{}&1.364352042889&{}:{}&0.803317803221&{}:{}&-1.167669846111&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{2c}}\)
\(P^-_{\mathbf{2c}}\)
\(P^+_{\mathbf{2c}}\)
\(Q_{\mathbf{2c}}\)
\(I^\prime_{\mathbf{2c}}\)
2c (130)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{2c}}&{}\approx{}&1.217668956838&{}:{}&0.297341324655&{}:{}&-0.515010281493&,\\B^\prime_{\mathbf{2c}}&{}\approx{}&29.136415405085&{}:{}&22.329336426955&{}:{}&-50.465751832040&,\\C^\prime_{\mathbf{2c}}&{}\approx{}&0.633974596216&{}:{}&0.633974596216&{}:{}&-0.267949192431&, \\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.240947612094&{}:{}&1.095987631846&{}:{}&-1.336935243939&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.031312801540&{}:{}&0.404818262014&{}:{}&-0.436131063554&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.499705569692&{}:{}&0.519909672579&{}:{}&-1.019615242271&, \\ A^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.744201338357&{}:{}&0.570335156968&{}:{}&-0.314536495325&,\\B^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.876437581058&{}:{}&0.214016387513&{}:{}&-0.090453968572&,\\C^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.308571806995&{}:{}&0.244885625659&{}:{}&-0.553457432654&, \\ A^*_{\mathbf{2c}}&{}\approx{}&0.000000000000&{}:{}&4.039058011260&{}:{}&-3.039058011260&,\\B^*_{\mathbf{2c}}&{}\approx{}&0.692992796309&{}:{}&0.000000000000&{}:{}&0.307007203691&,\\C^*_{\mathbf{2c}}&{}\approx{}&2.431851652578&{}:{}&-1.431851652578&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{2c}}}{B^\prime_{\mathbf{2c}}}{C^\prime_{\mathbf{2c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{2c}}}{B^{\prime\prime}_{\mathbf{2c}}}{C^{\prime\prime}_{\mathbf{2c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{2c}}}{B^{\prime\prime\prime}_{\mathbf{2c}}}{C^{\prime\prime\prime}_{\mathbf{2c}}}\)
\(\triangle{A^*_{\mathbf{2c}}}{B^*_{\mathbf{2c}}}{C^*_{\mathbf{2c}}}\)
2c (130)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{2c}}}}&{}\approx{}&0.079672367818&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{2c}}}}&{}\approx{}&7.807078978130&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{2c}}}}&{}\approx{}&0.169872981078&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.732050807569&{}:{}&3.732050807569&{}:{}&-6.464101615138&,\\ A^\prime_{\mathbf{2c}}&{}\approx{}&1.217668956838&{}:{}&0.297341324655&{}:{}&-0.515010281493&,\\B^\prime_{\mathbf{2c}}&{}\approx{}&29.136415405085&{}:{}&22.329336426955&{}:{}&-50.465751832040&,\\C^\prime_{\mathbf{2c}}&{}\approx{}&0.633974596216&{}:{}&0.633974596216&{}:{}&-0.267949192431&. \end{alignedat} \]
2c (130)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{2c}}&{}\approx{}&1.082502636859&{}:{}&0.375276056093&{}:{}&-0.457778692951&,\\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.240947612094&{}:{}&1.095987631846&{}:{}&-1.336935243939&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.031312801540&{}:{}&0.404818262014&{}:{}&-0.436131063554&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.499705569692&{}:{}&0.519909672579&{}:{}&-1.019615242271&. \end{alignedat} \]
2c (130)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{2c}}&{}\approx{}&0.922566064186&{}:{}&0.172648659120&{}:{}&-0.095214723306&,\\ A^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.744201338357&{}:{}&0.570335156968&{}:{}&-0.314536495325&,\\B^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.876437581058&{}:{}&0.214016387513&{}:{}&-0.090453968572&,\\C^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.308571806995&{}:{}&0.244885625659&{}:{}&-0.553457432654&. \end{alignedat} \]
2c (130)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{2c}}&{}\approx{}&1.225210460522&{}:{}&0.556075946659&{}:{}&-0.781286407181&,\\ A^\prime_{\mathbf{2c}}&{}\approx{}&1.217668956838&{}:{}&0.297341324655&{}:{}&-0.515010281493&,\\B^\prime_{\mathbf{2c}}&{}\approx{}&29.136415405085&{}:{}&22.329336426955&{}:{}&-50.465751832040&,\\C^\prime_{\mathbf{2c}}&{}\approx{}&0.633974596216&{}:{}&0.633974596216&{}:{}&-0.267949192431&,\\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.240947612094&{}:{}&1.095987631846&{}:{}&-1.336935243939&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.031312801540&{}:{}&0.404818262014&{}:{}&-0.436131063554&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.499705569692&{}:{}&0.519909672579&{}:{}&-1.019615242271&, \end{alignedat} \]
2c (130)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{2c}}&{}\approx{}&1.170650750933&{}:{}&-0.689268282684&{}:{}&0.518617531751&,\\ A^*_{\mathbf{2c}}&{}\approx{}&0.000000000000&{}:{}&4.039058011260&{}:{}&-3.039058011260&,\\B^*_{\mathbf{2c}}&{}\approx{}&0.692992796309&{}:{}&0.000000000000&{}:{}&0.307007203691&,\\C^*_{\mathbf{2c}}&{}\approx{}&2.431851652578&{}:{}&-1.431851652578&{}:{}&0.000000000000&. \end{alignedat} \]
2c (130)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{2c}}&{}\approx{}&1.364352042889&{}:{}&0.803317803221&{}:{}&-1.167669846111&,\\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.240947612094&{}:{}&1.095987631846&{}:{}&-1.336935243939&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.031312801540&{}:{}&0.404818262014&{}:{}&-0.436131063554&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.499705569692&{}:{}&0.519909672579&{}:{}&-1.019615242271&,\\ A^*_{\mathbf{2c}}&{}\approx{}&0.000000000000&{}:{}&4.039058011260&{}:{}&-3.039058011260&,\\B^*_{\mathbf{2c}}&{}\approx{}&0.692992796309&{}:{}&0.000000000000&{}:{}&0.307007203691&,\\C^*_{\mathbf{2c}}&{}\approx{}&2.431851652578&{}:{}&-1.431851652578&{}:{}&0.000000000000&. \end{alignedat} \]
2c (130)