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 > 3b (123)

3b (123)

Malfatti circles

3b (123)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.577350269190&{}:{}&-0.577350269190&{}:{}&1.000000000000&, \\ P_{\mathbf{3b}}&{}\approx{}&-0.000825825539&{}:{}&1.127104679846&{}:{}&-0.126278854307&, \\ P^-_{\mathbf{3b}}&{}\approx{}&0.094415041845&{}:{}&0.846335949968&{}:{}&0.059249008187&, \\ P^+_{\mathbf{3b}}&{}\approx{}&-0.142860372465&{}:{}&1.545820515123&{}:{}&-0.402960142658&, \\ Q_{\mathbf{3b}}&{}\approx{}&0.165692500245&{}:{}&1.640186660835&{}:{}&-0.805879161080&, \\ I^\prime_{\mathbf{3b}}&{}\approx{}&-0.050840841584&{}:{}&1.878238345509&{}:{}&-0.827397503925&, \end{alignedat} \]
\(I_{\mathbf{b}}\) Incenter
\(P_{\mathbf{3b}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{3b}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{3b}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{3b}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{3b}}\) Radical center of the Malfatti circles
3b (123)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3b}}&{}\approx{}&3.808060412490&{}:{}&3.835881858823&{}:{}&-6.643942271314&,\\B^\prime_{\mathbf{3b}}&{}\approx{}&-0.162155272456&{}:{}&1.443016443066&{}:{}&-0.280861170610&,\\C^\prime_{\mathbf{3b}}&{}\approx{}&-1.605171715523&{}:{}&1.605171715523&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.294554497996&{}:{}&1.457894466524&{}:{}&-0.163339968528&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.008207638122&{}:{}&2.263256179980&{}:{}&-1.255048541858&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.001128402279&{}:{}&1.540068003795&{}:{}&-0.538939601516&, \\ A^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.117343817645&{}:{}&1.044240225979&{}:{}&0.073103591666&,\\B^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.379529147818&{}:{}&0.382301963550&{}:{}&0.238168888632&,\\C^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.121665012933&{}:{}&1.090604550783&{}:{}&-0.212269563716&, \\ A^*_{\mathbf{3b}}&{}\approx{}&0.000000000000&{}:{}&1.965925826289&{}:{}&-0.965925826289&,\\B^*_{\mathbf{3b}}&{}\approx{}&-0.258819045103&{}:{}&0.000000000000&{}:{}&1.258819045103&,\\C^*_{\mathbf{3b}}&{}\approx{}&0.091751709536&{}:{}&0.908248290464&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3b}}}{B^\prime_{\mathbf{3b}}}{C^\prime_{\mathbf{3b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3b}}}{B^{\prime\prime}_{\mathbf{3b}}}{C^{\prime\prime}_{\mathbf{3b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3b}}}{B^{\prime\prime\prime}_{\mathbf{3b}}}{C^{\prime\prime\prime}_{\mathbf{3b}}}\)
\(\triangle{A^*_{\mathbf{3b}}}{B^*_{\mathbf{3b}}}{C^*_{\mathbf{3b}}}\)
3b (123)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3b}}}}&{}\approx{}&-6.643942271314&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3b}}}}&{}\approx{}&-0.280861170610&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3b}}}}&{}\approx{}&-2.780238966158&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.577350269190&{}:{}&-0.577350269190&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{3b}}&{}\approx{}&3.808060412490&{}:{}&3.835881858823&{}:{}&-6.643942271314&,\\B^\prime_{\mathbf{3b}}&{}\approx{}&-0.162155272456&{}:{}&1.443016443066&{}:{}&-0.280861170610&,\\C^\prime_{\mathbf{3b}}&{}\approx{}&-1.605171715523&{}:{}&1.605171715523&{}:{}&1.000000000000&. \end{alignedat} \]
3b (123)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3b}}&{}\approx{}&-0.000825825539&{}:{}&1.127104679846&{}:{}&-0.126278854307&,\\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.294554497996&{}:{}&1.457894466524&{}:{}&-0.163339968528&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.008207638122&{}:{}&2.263256179980&{}:{}&-1.255048541858&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.001128402279&{}:{}&1.540068003795&{}:{}&-0.538939601516&. \end{alignedat} \]
3b (123)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3b}}&{}\approx{}&0.094415041845&{}:{}&0.846335949968&{}:{}&0.059249008187&,\\ A^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.117343817645&{}:{}&1.044240225979&{}:{}&0.073103591666&,\\B^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.379529147818&{}:{}&0.382301963550&{}:{}&0.238168888632&,\\C^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.121665012933&{}:{}&1.090604550783&{}:{}&-0.212269563716&. \end{alignedat} \]
3b (123)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3b}}&{}\approx{}&-0.142860372465&{}:{}&1.545820515123&{}:{}&-0.402960142658&,\\ A^\prime_{\mathbf{3b}}&{}\approx{}&3.808060412490&{}:{}&3.835881858823&{}:{}&-6.643942271314&,\\B^\prime_{\mathbf{3b}}&{}\approx{}&-0.162155272456&{}:{}&1.443016443066&{}:{}&-0.280861170610&,\\C^\prime_{\mathbf{3b}}&{}\approx{}&-1.605171715523&{}:{}&1.605171715523&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.294554497996&{}:{}&1.457894466524&{}:{}&-0.163339968528&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.008207638122&{}:{}&2.263256179980&{}:{}&-1.255048541858&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.001128402279&{}:{}&1.540068003795&{}:{}&-0.538939601516&, \end{alignedat} \]
3b (123)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3b}}&{}\approx{}&0.165692500245&{}:{}&1.640186660835&{}:{}&-0.805879161080&,\\ A^*_{\mathbf{3b}}&{}\approx{}&0.000000000000&{}:{}&1.965925826289&{}:{}&-0.965925826289&,\\B^*_{\mathbf{3b}}&{}\approx{}&-0.258819045103&{}:{}&0.000000000000&{}:{}&1.258819045103&,\\C^*_{\mathbf{3b}}&{}\approx{}&0.091751709536&{}:{}&0.908248290464&{}:{}&0.000000000000&. \end{alignedat} \]
3b (123)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3b}}&{}\approx{}&-0.050840841584&{}:{}&1.878238345509&{}:{}&-0.827397503925&,\\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.294554497996&{}:{}&1.457894466524&{}:{}&-0.163339968528&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.008207638122&{}:{}&2.263256179980&{}:{}&-1.255048541858&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.001128402279&{}:{}&1.540068003795&{}:{}&-0.538939601516&,\\ A^*_{\mathbf{3b}}&{}\approx{}&0.000000000000&{}:{}&1.965925826289&{}:{}&-0.965925826289&,\\B^*_{\mathbf{3b}}&{}\approx{}&-0.258819045103&{}:{}&0.000000000000&{}:{}&1.258819045103&,\\C^*_{\mathbf{3b}}&{}\approx{}&0.091751709536&{}:{}&0.908248290464&{}:{}&0.000000000000&. \end{alignedat} \]
3b (123)