Derousseau's Generalization of the Malfatti circles

Isosceles Triangle with 135° Top Angle

\(A=22.5\degree\), \(B=22.5\degree\), \(C=135\degree\).


[Top] > Isosceles Triangle with 135° Top Angle > 0c (110)

0c(110)

Malfatti circles

0c (110)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&6.568535592272&{}:{}&6.568535592272&{}:{}&-12.137071184544&, \\ P_{\mathbf{c}}&{}\approx{}&1.613914110072&{}:{}&1.613914110072&{}:{}&-2.227828220144&, \\ P^-_{\mathbf{c}}&{}\approx{}&1.130766460211&{}:{}&1.130766460211&{}:{}&-1.261532920422&, \\ P^+_{\mathbf{c}}&{}\approx{}&2.018211926313&{}:{}&2.018211926313&{}:{}&-3.036423852626&, \\ Q_{\mathbf{c}}&{}\approx{}&0.843296904551&{}:{}&0.843296904551&{}:{}&-0.686593809102&, \\ I^\prime_{\mathbf{c}}&{}\approx{}&2.481806171257&{}:{}&2.481806171257&{}:{}&-3.963612342513&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{c}}\)
\(P^-_{\mathbf{c}}\)
\(P^+_{\mathbf{c}}\)
\(Q_{\mathbf{c}}\)
\(I^\prime_{\mathbf{c}}\)
0c (110)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{c}}&{}\approx{}&1.923879532511&{}:{}&1.089790213552&{}:{}&-2.013669746063&,\\B^\prime_{\mathbf{c}}&{}\approx{}&1.089790213552&{}:{}&1.923879532511&{}:{}&-2.013669746063&,\\C^\prime_{\mathbf{c}}&{}\approx{}&32.809596275628&{}:{}&32.809596275628&{}:{}&-64.619192551255&, \\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&2.109511742531&{}:{}&2.916786936777&{}:{}&-4.026298679308&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&2.916786936777&{}:{}&2.109511742531&{}:{}&-4.026298679308&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.506834873031&{}:{}&1.506834873031&{}:{}&-2.013669746063&, \\ A^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&1.256524894379&{}:{}&2.218227413245&{}:{}&-2.474752307623&,\\B^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&2.218227413245&{}:{}&1.256524894379&{}:{}&-2.474752307623&,\\C^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&1.048956292708&{}:{}&1.048956292708&{}:{}&-1.097912585416&, \\ A^*_{\mathbf{c}}&{}\approx{}&0.000000000000&{}:{}&5.381494871788&{}:{}&-4.381494871788&,\\B^*_{\mathbf{c}}&{}\approx{}&5.381494871788&{}:{}&0.000000000000&{}:{}&-4.381494871788&,\\C^*_{\mathbf{c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{c}}}{B^\prime_{\mathbf{c}}}{C^\prime_{\mathbf{c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{c}}}{B^{\prime\prime}_{\mathbf{c}}}{C^{\prime\prime}_{\mathbf{c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{c}}}{B^{\prime\prime\prime}_{\mathbf{c}}}{C^{\prime\prime\prime}_{\mathbf{c}}}\)
\(\triangle{A^*_{\mathbf{c}}}{B^*_{\mathbf{c}}}{C^*_{\mathbf{c}}}\)
0c (110)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{c}}}}&{}\approx{}&0.165910681040&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{c}}}}&{}\approx{}&0.165910681040&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{c}}}}&{}\approx{}&4.994963613233&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&6.568535592272&{}:{}&6.568535592272&{}:{}&-12.137071184544&,\\ A^\prime_{\mathbf{c}}&{}\approx{}&1.923879532511&{}:{}&1.089790213552&{}:{}&-2.013669746063&,\\B^\prime_{\mathbf{c}}&{}\approx{}&1.089790213552&{}:{}&1.923879532511&{}:{}&-2.013669746063&,\\C^\prime_{\mathbf{c}}&{}\approx{}&32.809596275628&{}:{}&32.809596275628&{}:{}&-64.619192551255&. \end{alignedat} \]
0c (110)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{c}}&{}\approx{}&1.613914110072&{}:{}&1.613914110072&{}:{}&-2.227828220144&,\\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&2.109511742531&{}:{}&2.916786936777&{}:{}&-4.026298679308&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&2.916786936777&{}:{}&2.109511742531&{}:{}&-4.026298679308&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.506834873031&{}:{}&1.506834873031&{}:{}&-2.013669746063&. \end{alignedat} \]
0c (110)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{c}}&{}\approx{}&1.130766460211&{}:{}&1.130766460211&{}:{}&-1.261532920422&,\\ A^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&1.256524894379&{}:{}&2.218227413245&{}:{}&-2.474752307623&,\\B^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&2.218227413245&{}:{}&1.256524894379&{}:{}&-2.474752307623&,\\C^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&1.048956292708&{}:{}&1.048956292708&{}:{}&-1.097912585416&. \end{alignedat} \]
0c (110)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{c}}&{}\approx{}&2.018211926313&{}:{}&2.018211926313&{}:{}&-3.036423852626&,\\ A^\prime_{\mathbf{c}}&{}\approx{}&1.923879532511&{}:{}&1.089790213552&{}:{}&-2.013669746063&,\\B^\prime_{\mathbf{c}}&{}\approx{}&1.089790213552&{}:{}&1.923879532511&{}:{}&-2.013669746063&,\\C^\prime_{\mathbf{c}}&{}\approx{}&32.809596275628&{}:{}&32.809596275628&{}:{}&-64.619192551255&,\\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&2.109511742531&{}:{}&2.916786936777&{}:{}&-4.026298679308&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&2.916786936777&{}:{}&2.109511742531&{}:{}&-4.026298679308&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.506834873031&{}:{}&1.506834873031&{}:{}&-2.013669746063&, \end{alignedat} \]
0c (110)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{c}}&{}\approx{}&0.843296904551&{}:{}&0.843296904551&{}:{}&-0.686593809102&,\\ A^*_{\mathbf{c}}&{}\approx{}&0.000000000000&{}:{}&5.381494871788&{}:{}&-4.381494871788&,\\B^*_{\mathbf{c}}&{}\approx{}&5.381494871788&{}:{}&0.000000000000&{}:{}&-4.381494871788&,\\C^*_{\mathbf{c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
0c (110)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{c}}&{}\approx{}&2.481806171257&{}:{}&2.481806171257&{}:{}&-3.963612342513&,\\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&2.109511742531&{}:{}&2.916786936777&{}:{}&-4.026298679308&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&2.916786936777&{}:{}&2.109511742531&{}:{}&-4.026298679308&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.506834873031&{}:{}&1.506834873031&{}:{}&-2.013669746063&,\\ A^*_{\mathbf{c}}&{}\approx{}&0.000000000000&{}:{}&5.381494871788&{}:{}&-4.381494871788&,\\B^*_{\mathbf{c}}&{}\approx{}&5.381494871788&{}:{}&0.000000000000&{}:{}&-4.381494871788&,\\C^*_{\mathbf{c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
0c (110)