Derousseau's Generalization of the Malfatti circles

Isosceles Triangle with 90° Top Angle

\(A=45\degree\), \(B=45\degree\), \(C=90\degree\).


[Top] > Isosceles Triangle with 90° Top Angle > 0c (110)

0c(110)

Malfatti circles

0c (110)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.707106781187&{}:{}&1.707106781187&{}:{}&-2.414213562373&, \\ P_{\mathbf{c}}&{}\approx{}&0.932633521216&{}:{}&0.932633521216&{}:{}&-0.865267042431&, \\ P^-_{\mathbf{c}}&{}\approx{}&0.785098763416&{}:{}&0.785098763416&{}:{}&-0.570197526832&, \\ P^+_{\mathbf{c}}&{}\approx{}&1.039465830724&{}:{}&1.039465830724&{}:{}&-1.078931661448&, \\ Q_{\mathbf{c}}&{}\approx{}&0.724592737729&{}:{}&0.724592737729&{}:{}&-0.449185475457&, \\ I^\prime_{\mathbf{c}}&{}\approx{}&1.170209910570&{}:{}&1.170209910570&{}:{}&-1.340419821139&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{c}}\)
\(P^-_{\mathbf{c}}\)
\(P^+_{\mathbf{c}}\)
\(Q_{\mathbf{c}}\)
\(I^\prime_{\mathbf{c}}\)
0c (110)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{c}}&{}\approx{}&1.207106781187&{}:{}&0.500000000000&{}:{}&-0.707106781187&,\\B^\prime_{\mathbf{c}}&{}\approx{}&0.500000000000&{}:{}&1.207106781187&{}:{}&-0.707106781187&,\\C^\prime_{\mathbf{c}}&{}\approx{}&4.054865846209&{}:{}&4.054865846209&{}:{}&-7.109731692418&, \\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.890227282892&{}:{}&1.519713031423&{}:{}&-1.409940314315&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.519713031423&{}:{}&0.890227282892&{}:{}&-1.409940314315&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.853553390593&{}:{}&0.853553390593&{}:{}&-0.707106781187&, \\ A^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.602108277599&{}:{}&1.453617969797&{}:{}&-1.055726247397&,\\B^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&1.453617969797&{}:{}&0.602108277599&{}:{}&-1.055726247397&,\\C^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.707106781187&{}:{}&0.707106781187&{}:{}&-0.414213562373&, \\ A^*_{\mathbf{c}}&{}\approx{}&0.000000000000&{}:{}&2.630986313698&{}:{}&-1.630986313698&,\\B^*_{\mathbf{c}}&{}\approx{}&2.630986313698&{}:{}&0.000000000000&{}:{}&-1.630986313698&,\\C^*_{\mathbf{c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{c}}}{B^\prime_{\mathbf{c}}}{C^\prime_{\mathbf{c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{c}}}{B^{\prime\prime}_{\mathbf{c}}}{C^{\prime\prime}_{\mathbf{c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{c}}}{B^{\prime\prime\prime}_{\mathbf{c}}}{C^{\prime\prime\prime}_{\mathbf{c}}}\)
\(\triangle{A^*_{\mathbf{c}}}{B^*_{\mathbf{c}}}{C^*_{\mathbf{c}}}\)
0c (110)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{c}}}}&{}\approx{}&0.292893218813&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{c}}}}&{}\approx{}&0.292893218813&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{c}}}}&{}\approx{}&2.375285419106&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.707106781187&{}:{}&1.707106781187&{}:{}&-2.414213562373&,\\ A^\prime_{\mathbf{c}}&{}\approx{}&1.207106781187&{}:{}&0.500000000000&{}:{}&-0.707106781187&,\\B^\prime_{\mathbf{c}}&{}\approx{}&0.500000000000&{}:{}&1.207106781187&{}:{}&-0.707106781187&,\\C^\prime_{\mathbf{c}}&{}\approx{}&4.054865846209&{}:{}&4.054865846209&{}:{}&-7.109731692418&. \end{alignedat} \]
0c (110)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{c}}&{}\approx{}&0.932633521216&{}:{}&0.932633521216&{}:{}&-0.865267042431&,\\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.890227282892&{}:{}&1.519713031423&{}:{}&-1.409940314315&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.519713031423&{}:{}&0.890227282892&{}:{}&-1.409940314315&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.853553390593&{}:{}&0.853553390593&{}:{}&-0.707106781187&. \end{alignedat} \]
0c (110)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{c}}&{}\approx{}&0.785098763416&{}:{}&0.785098763416&{}:{}&-0.570197526832&,\\ A^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.602108277599&{}:{}&1.453617969797&{}:{}&-1.055726247397&,\\B^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&1.453617969797&{}:{}&0.602108277599&{}:{}&-1.055726247397&,\\C^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.707106781187&{}:{}&0.707106781187&{}:{}&-0.414213562373&. \end{alignedat} \]
0c (110)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{c}}&{}\approx{}&1.039465830724&{}:{}&1.039465830724&{}:{}&-1.078931661448&,\\ A^\prime_{\mathbf{c}}&{}\approx{}&1.207106781187&{}:{}&0.500000000000&{}:{}&-0.707106781187&,\\B^\prime_{\mathbf{c}}&{}\approx{}&0.500000000000&{}:{}&1.207106781187&{}:{}&-0.707106781187&,\\C^\prime_{\mathbf{c}}&{}\approx{}&4.054865846209&{}:{}&4.054865846209&{}:{}&-7.109731692418&,\\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.890227282892&{}:{}&1.519713031423&{}:{}&-1.409940314315&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.519713031423&{}:{}&0.890227282892&{}:{}&-1.409940314315&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.853553390593&{}:{}&0.853553390593&{}:{}&-0.707106781187&, \end{alignedat} \]
0c (110)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{c}}&{}\approx{}&0.724592737729&{}:{}&0.724592737729&{}:{}&-0.449185475457&,\\ A^*_{\mathbf{c}}&{}\approx{}&0.000000000000&{}:{}&2.630986313698&{}:{}&-1.630986313698&,\\B^*_{\mathbf{c}}&{}\approx{}&2.630986313698&{}:{}&0.000000000000&{}:{}&-1.630986313698&,\\C^*_{\mathbf{c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
0c (110)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{c}}&{}\approx{}&1.170209910570&{}:{}&1.170209910570&{}:{}&-1.340419821139&,\\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.890227282892&{}:{}&1.519713031423&{}:{}&-1.409940314315&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.519713031423&{}:{}&0.890227282892&{}:{}&-1.409940314315&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.853553390593&{}:{}&0.853553390593&{}:{}&-0.707106781187&,\\ A^*_{\mathbf{c}}&{}\approx{}&0.000000000000&{}:{}&2.630986313698&{}:{}&-1.630986313698&,\\B^*_{\mathbf{c}}&{}\approx{}&2.630986313698&{}:{}&0.000000000000&{}:{}&-1.630986313698&,\\C^*_{\mathbf{c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
0c (110)