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 > 5a (213)

5a(213)

Malfatti circles

5a (213)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.577350269190&{}:{}&0.577350269190&{}:{}&1.000000000000&, \\ P_{\mathbf{5a}}&{}\approx{}&1.127104679846&{}:{}&-0.000825825539&{}:{}&-0.126278854307&, \\ P^-_{\mathbf{5a}}&{}\approx{}&0.846335949968&{}:{}&0.094415041845&{}:{}&0.059249008187&, \\ P^+_{\mathbf{5a}}&{}\approx{}&1.545820515123&{}:{}&-0.142860372465&{}:{}&-0.402960142658&, \\ Q_{\mathbf{5a}}&{}\approx{}&1.640186660835&{}:{}&0.165692500245&{}:{}&-0.805879161080&, \\ I^\prime_{\mathbf{5a}}&{}\approx{}&1.878238345509&{}:{}&-0.050840841584&{}:{}&-0.827397503925&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{5a}}\)
\(P^-_{\mathbf{5a}}\)
\(P^+_{\mathbf{5a}}\)
\(Q_{\mathbf{5a}}\)
\(I^\prime_{\mathbf{5a}}\)
5a (213)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5a}}&{}\approx{}&1.443016443066&{}:{}&-0.162155272456&{}:{}&-0.280861170610&,\\B^\prime_{\mathbf{5a}}&{}\approx{}&3.835881858823&{}:{}&3.808060412490&{}:{}&-6.643942271314&,\\C^\prime_{\mathbf{5a}}&{}\approx{}&1.605171715523&{}:{}&-1.605171715523&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&2.263256179980&{}:{}&-0.008207638122&{}:{}&-1.255048541858&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.457894466524&{}:{}&-0.294554497996&{}:{}&-0.163339968528&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.540068003795&{}:{}&-0.001128402279&{}:{}&-0.538939601516&, \\ A^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&0.382301963550&{}:{}&0.379529147818&{}:{}&0.238168888632&,\\B^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.044240225979&{}:{}&-0.117343817645&{}:{}&0.073103591666&,\\C^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.090604550783&{}:{}&0.121665012933&{}:{}&-0.212269563716&, \\ A^*_{\mathbf{5a}}&{}\approx{}&0.000000000000&{}:{}&-0.258819045103&{}:{}&1.258819045103&,\\B^*_{\mathbf{5a}}&{}\approx{}&1.965925826289&{}:{}&0.000000000000&{}:{}&-0.965925826289&,\\C^*_{\mathbf{5a}}&{}\approx{}&0.908248290464&{}:{}&0.091751709536&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5a}}}{B^\prime_{\mathbf{5a}}}{C^\prime_{\mathbf{5a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5a}}}{B^{\prime\prime}_{\mathbf{5a}}}{C^{\prime\prime}_{\mathbf{5a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5a}}}{B^{\prime\prime\prime}_{\mathbf{5a}}}{C^{\prime\prime\prime}_{\mathbf{5a}}}\)
\(\triangle{A^*_{\mathbf{5a}}}{B^*_{\mathbf{5a}}}{C^*_{\mathbf{5a}}}\)
5a (213)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5a}}}}&{}\approx{}&-0.280861170610&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5a}}}}&{}\approx{}&-6.643942271314&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5a}}}}&{}\approx{}&-2.780238966158&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.577350269190&{}:{}&0.577350269190&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{5a}}&{}\approx{}&1.443016443066&{}:{}&-0.162155272456&{}:{}&-0.280861170610&,\\B^\prime_{\mathbf{5a}}&{}\approx{}&3.835881858823&{}:{}&3.808060412490&{}:{}&-6.643942271314&,\\C^\prime_{\mathbf{5a}}&{}\approx{}&1.605171715523&{}:{}&-1.605171715523&{}:{}&1.000000000000&. \end{alignedat} \]
5a (213)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5a}}&{}\approx{}&1.127104679846&{}:{}&-0.000825825539&{}:{}&-0.126278854307&,\\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&2.263256179980&{}:{}&-0.008207638122&{}:{}&-1.255048541858&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.457894466524&{}:{}&-0.294554497996&{}:{}&-0.163339968528&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.540068003795&{}:{}&-0.001128402279&{}:{}&-0.538939601516&. \end{alignedat} \]
5a (213)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5a}}&{}\approx{}&0.846335949968&{}:{}&0.094415041845&{}:{}&0.059249008187&,\\ A^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&0.382301963550&{}:{}&0.379529147818&{}:{}&0.238168888632&,\\B^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.044240225979&{}:{}&-0.117343817645&{}:{}&0.073103591666&,\\C^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.090604550783&{}:{}&0.121665012933&{}:{}&-0.212269563716&. \end{alignedat} \]
5a (213)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5a}}&{}\approx{}&1.545820515123&{}:{}&-0.142860372465&{}:{}&-0.402960142658&,\\ A^\prime_{\mathbf{5a}}&{}\approx{}&1.443016443066&{}:{}&-0.162155272456&{}:{}&-0.280861170610&,\\B^\prime_{\mathbf{5a}}&{}\approx{}&3.835881858823&{}:{}&3.808060412490&{}:{}&-6.643942271314&,\\C^\prime_{\mathbf{5a}}&{}\approx{}&1.605171715523&{}:{}&-1.605171715523&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&2.263256179980&{}:{}&-0.008207638122&{}:{}&-1.255048541858&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.457894466524&{}:{}&-0.294554497996&{}:{}&-0.163339968528&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.540068003795&{}:{}&-0.001128402279&{}:{}&-0.538939601516&, \end{alignedat} \]
5a (213)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5a}}&{}\approx{}&1.640186660835&{}:{}&0.165692500245&{}:{}&-0.805879161080&,\\ A^*_{\mathbf{5a}}&{}\approx{}&0.000000000000&{}:{}&-0.258819045103&{}:{}&1.258819045103&,\\B^*_{\mathbf{5a}}&{}\approx{}&1.965925826289&{}:{}&0.000000000000&{}:{}&-0.965925826289&,\\C^*_{\mathbf{5a}}&{}\approx{}&0.908248290464&{}:{}&0.091751709536&{}:{}&0.000000000000&. \end{alignedat} \]
5a (213)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5a}}&{}\approx{}&1.878238345509&{}:{}&-0.050840841584&{}:{}&-0.827397503925&,\\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&2.263256179980&{}:{}&-0.008207638122&{}:{}&-1.255048541858&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.457894466524&{}:{}&-0.294554497996&{}:{}&-0.163339968528&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.540068003795&{}:{}&-0.001128402279&{}:{}&-0.538939601516&,\\ A^*_{\mathbf{5a}}&{}\approx{}&0.000000000000&{}:{}&-0.258819045103&{}:{}&1.258819045103&,\\B^*_{\mathbf{5a}}&{}\approx{}&1.965925826289&{}:{}&0.000000000000&{}:{}&-0.965925826289&,\\C^*_{\mathbf{5a}}&{}\approx{}&0.908248290464&{}:{}&0.091751709536&{}:{}&0.000000000000&. \end{alignedat} \]
5a (213)