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 > 4c (310)

4c(310)

Malfatti circles

4c (310)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.732050807569&{}:{}&3.732050807569&{}:{}&-6.464101615138&, \\ P_{\mathbf{4c}}&{}\approx{}&0.375276056093&{}:{}&1.082502636859&{}:{}&-0.457778692951&, \\ P^-_{\mathbf{4c}}&{}\approx{}&0.172648659120&{}:{}&0.922566064186&{}:{}&-0.095214723306&, \\ P^+_{\mathbf{4c}}&{}\approx{}&0.556075946659&{}:{}&1.225210460522&{}:{}&-0.781286407181&, \\ Q_{\mathbf{4c}}&{}\approx{}&-0.689268282684&{}:{}&1.170650750933&{}:{}&0.518617531751&, \\ I^\prime_{\mathbf{4c}}&{}\approx{}&0.803317803221&{}:{}&1.364352042889&{}:{}&-1.167669846111&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{4c}}\)
\(P^-_{\mathbf{4c}}\)
\(P^+_{\mathbf{4c}}\)
\(Q_{\mathbf{4c}}\)
\(I^\prime_{\mathbf{4c}}\)
4c (310)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4c}}&{}\approx{}&22.329336426955&{}:{}&29.136415405085&{}:{}&-50.465751832040&,\\B^\prime_{\mathbf{4c}}&{}\approx{}&0.297341324655&{}:{}&1.217668956838&{}:{}&-0.515010281493&,\\C^\prime_{\mathbf{4c}}&{}\approx{}&0.633974596216&{}:{}&0.633974596216&{}:{}&-0.267949192431&, \\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.404818262014&{}:{}&1.031312801540&{}:{}&-0.436131063554&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&1.095987631846&{}:{}&1.240947612094&{}:{}&-1.336935243939&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.519909672579&{}:{}&1.499705569692&{}:{}&-1.019615242271&, \\ A^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.214016387513&{}:{}&0.876437581058&{}:{}&-0.090453968572&,\\B^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.570335156968&{}:{}&0.744201338357&{}:{}&-0.314536495325&,\\C^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.244885625659&{}:{}&1.308571806995&{}:{}&-0.553457432654&, \\ A^*_{\mathbf{4c}}&{}\approx{}&0.000000000000&{}:{}&0.692992796309&{}:{}&0.307007203691&,\\B^*_{\mathbf{4c}}&{}\approx{}&4.039058011260&{}:{}&0.000000000000&{}:{}&-3.039058011260&,\\C^*_{\mathbf{4c}}&{}\approx{}&-1.431851652578&{}:{}&2.431851652578&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4c}}}{B^\prime_{\mathbf{4c}}}{C^\prime_{\mathbf{4c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4c}}}{B^{\prime\prime}_{\mathbf{4c}}}{C^{\prime\prime}_{\mathbf{4c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4c}}}{B^{\prime\prime\prime}_{\mathbf{4c}}}{C^{\prime\prime\prime}_{\mathbf{4c}}}\)
\(\triangle{A^*_{\mathbf{4c}}}{B^*_{\mathbf{4c}}}{C^*_{\mathbf{4c}}}\)
4c (310)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4c}}}}&{}\approx{}&7.807078978130&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4c}}}}&{}\approx{}&0.079672367818&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4c}}}}&{}\approx{}&0.169872981078&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.732050807569&{}:{}&3.732050807569&{}:{}&-6.464101615138&,\\ A^\prime_{\mathbf{4c}}&{}\approx{}&22.329336426955&{}:{}&29.136415405085&{}:{}&-50.465751832040&,\\B^\prime_{\mathbf{4c}}&{}\approx{}&0.297341324655&{}:{}&1.217668956838&{}:{}&-0.515010281493&,\\C^\prime_{\mathbf{4c}}&{}\approx{}&0.633974596216&{}:{}&0.633974596216&{}:{}&-0.267949192431&. \end{alignedat} \]
4c (310)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4c}}&{}\approx{}&0.375276056093&{}:{}&1.082502636859&{}:{}&-0.457778692951&,\\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.404818262014&{}:{}&1.031312801540&{}:{}&-0.436131063554&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&1.095987631846&{}:{}&1.240947612094&{}:{}&-1.336935243939&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.519909672579&{}:{}&1.499705569692&{}:{}&-1.019615242271&. \end{alignedat} \]
4c (310)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4c}}&{}\approx{}&0.172648659120&{}:{}&0.922566064186&{}:{}&-0.095214723306&,\\ A^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.214016387513&{}:{}&0.876437581058&{}:{}&-0.090453968572&,\\B^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.570335156968&{}:{}&0.744201338357&{}:{}&-0.314536495325&,\\C^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.244885625659&{}:{}&1.308571806995&{}:{}&-0.553457432654&. \end{alignedat} \]
4c (310)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4c}}&{}\approx{}&0.556075946659&{}:{}&1.225210460522&{}:{}&-0.781286407181&,\\ A^\prime_{\mathbf{4c}}&{}\approx{}&22.329336426955&{}:{}&29.136415405085&{}:{}&-50.465751832040&,\\B^\prime_{\mathbf{4c}}&{}\approx{}&0.297341324655&{}:{}&1.217668956838&{}:{}&-0.515010281493&,\\C^\prime_{\mathbf{4c}}&{}\approx{}&0.633974596216&{}:{}&0.633974596216&{}:{}&-0.267949192431&,\\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.404818262014&{}:{}&1.031312801540&{}:{}&-0.436131063554&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&1.095987631846&{}:{}&1.240947612094&{}:{}&-1.336935243939&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.519909672579&{}:{}&1.499705569692&{}:{}&-1.019615242271&, \end{alignedat} \]
4c (310)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4c}}&{}\approx{}&-0.689268282684&{}:{}&1.170650750933&{}:{}&0.518617531751&,\\ A^*_{\mathbf{4c}}&{}\approx{}&0.000000000000&{}:{}&0.692992796309&{}:{}&0.307007203691&,\\B^*_{\mathbf{4c}}&{}\approx{}&4.039058011260&{}:{}&0.000000000000&{}:{}&-3.039058011260&,\\C^*_{\mathbf{4c}}&{}\approx{}&-1.431851652578&{}:{}&2.431851652578&{}:{}&0.000000000000&. \end{alignedat} \]
4c (310)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4c}}&{}\approx{}&0.803317803221&{}:{}&1.364352042889&{}:{}&-1.167669846111&,\\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.404818262014&{}:{}&1.031312801540&{}:{}&-0.436131063554&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&1.095987631846&{}:{}&1.240947612094&{}:{}&-1.336935243939&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.519909672579&{}:{}&1.499705569692&{}:{}&-1.019615242271&,\\ A^*_{\mathbf{4c}}&{}\approx{}&0.000000000000&{}:{}&0.692992796309&{}:{}&0.307007203691&,\\B^*_{\mathbf{4c}}&{}\approx{}&4.039058011260&{}:{}&0.000000000000&{}:{}&-3.039058011260&,\\C^*_{\mathbf{4c}}&{}\approx{}&-1.431851652578&{}:{}&2.431851652578&{}:{}&0.000000000000&. \end{alignedat} \]
4c (310)