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 > 1a (013)

1a(013)

Malfatti circles

1a (013)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.707106781187&{}:{}&0.707106781187&{}:{}&1.000000000000&, \\ P_{\mathbf{1a}}&{}\approx{}&-0.095157655782&{}:{}&0.924237568416&{}:{}&0.170920087366&, \\ P^-_{\mathbf{1a}}&{}\approx{}&-0.009458955395&{}:{}&0.954645041042&{}:{}&0.054813914352&, \\ P^+_{\mathbf{1a}}&{}\approx{}&-0.162105351339&{}:{}&0.900483296637&{}:{}&0.261622054702&, \\ Q_{\mathbf{1a}}&{}\approx{}&0.155242065312&{}:{}&1.168032800651&{}:{}&-0.323274865964&, \\ I^\prime_{\mathbf{1a}}&{}\approx{}&-0.269518956564&{}:{}&0.839961263543&{}:{}&0.429557693021&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{1a}}\)
\(P^-_{\mathbf{1a}}\)
\(P^+_{\mathbf{1a}}\)
\(Q_{\mathbf{1a}}\)
\(I^\prime_{\mathbf{1a}}\)
1a (013)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1a}}&{}\approx{}&-0.041196100146&{}:{}&0.431277545771&{}:{}&0.609918554376&,\\B^\prime_{\mathbf{1a}}&{}\approx{}&-0.099456183690&{}:{}&0.958803899854&{}:{}&0.140652283836&,\\C^\prime_{\mathbf{1a}}&{}\approx{}&-1.806562964876&{}:{}&1.806562964876&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.188533109142&{}:{}&1.003040014354&{}:{}&0.185493094788&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.381418330097&{}:{}&0.696323118707&{}:{}&0.685095211389&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.088538608173&{}:{}&0.859948758263&{}:{}&0.228589849910&, \\ A^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.108766554612&{}:{}&1.048560208790&{}:{}&0.060206345821&,\\B^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.065514420258&{}:{}&0.685863425930&{}:{}&0.379650994327&,\\C^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.008716097378&{}:{}&0.879672098223&{}:{}&0.129043999154&, \\ A^*_{\mathbf{1a}}&{}\approx{}&0.000000000000&{}:{}&1.382683432365&{}:{}&-0.382683432365&,\\B^*_{\mathbf{1a}}&{}\approx{}&-0.923879532511&{}:{}&0.000000000000&{}:{}&1.923879532511&,\\C^*_{\mathbf{1a}}&{}\approx{}&0.117316567635&{}:{}&0.882683432365&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1a}}}{B^\prime_{\mathbf{1a}}}{C^\prime_{\mathbf{1a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1a}}}{B^{\prime\prime}_{\mathbf{1a}}}{C^{\prime\prime}_{\mathbf{1a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1a}}}{B^{\prime\prime\prime}_{\mathbf{1a}}}{C^{\prime\prime\prime}_{\mathbf{1a}}}\)
\(\triangle{A^*_{\mathbf{1a}}}{B^*_{\mathbf{1a}}}{C^*_{\mathbf{1a}}}\)
1a (013)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1a}}}}&{}\approx{}&0.609918554376&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1a}}}}&{}\approx{}&0.140652283836&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1a}}}}&{}\approx{}&2.554865846209&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.707106781187&{}:{}&0.707106781187&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{1a}}&{}\approx{}&-0.041196100146&{}:{}&0.431277545771&{}:{}&0.609918554376&,\\B^\prime_{\mathbf{1a}}&{}\approx{}&-0.099456183690&{}:{}&0.958803899854&{}:{}&0.140652283836&,\\C^\prime_{\mathbf{1a}}&{}\approx{}&-1.806562964876&{}:{}&1.806562964876&{}:{}&1.000000000000&. \end{alignedat} \]
1a (013)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1a}}&{}\approx{}&-0.095157655782&{}:{}&0.924237568416&{}:{}&0.170920087366&,\\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.188533109142&{}:{}&1.003040014354&{}:{}&0.185493094788&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.381418330097&{}:{}&0.696323118707&{}:{}&0.685095211389&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.088538608173&{}:{}&0.859948758263&{}:{}&0.228589849910&. \end{alignedat} \]
1a (013)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1a}}&{}\approx{}&-0.009458955395&{}:{}&0.954645041042&{}:{}&0.054813914352&,\\ A^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.108766554612&{}:{}&1.048560208790&{}:{}&0.060206345821&,\\B^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.065514420258&{}:{}&0.685863425930&{}:{}&0.379650994327&,\\C^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.008716097378&{}:{}&0.879672098223&{}:{}&0.129043999154&. \end{alignedat} \]
1a (013)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1a}}&{}\approx{}&-0.162105351339&{}:{}&0.900483296637&{}:{}&0.261622054702&,\\ A^\prime_{\mathbf{1a}}&{}\approx{}&-0.041196100146&{}:{}&0.431277545771&{}:{}&0.609918554376&,\\B^\prime_{\mathbf{1a}}&{}\approx{}&-0.099456183690&{}:{}&0.958803899854&{}:{}&0.140652283836&,\\C^\prime_{\mathbf{1a}}&{}\approx{}&-1.806562964876&{}:{}&1.806562964876&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.188533109142&{}:{}&1.003040014354&{}:{}&0.185493094788&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.381418330097&{}:{}&0.696323118707&{}:{}&0.685095211389&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.088538608173&{}:{}&0.859948758263&{}:{}&0.228589849910&, \end{alignedat} \]
1a (013)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1a}}&{}\approx{}&0.155242065312&{}:{}&1.168032800651&{}:{}&-0.323274865964&,\\ A^*_{\mathbf{1a}}&{}\approx{}&0.000000000000&{}:{}&1.382683432365&{}:{}&-0.382683432365&,\\B^*_{\mathbf{1a}}&{}\approx{}&-0.923879532511&{}:{}&0.000000000000&{}:{}&1.923879532511&,\\C^*_{\mathbf{1a}}&{}\approx{}&0.117316567635&{}:{}&0.882683432365&{}:{}&0.000000000000&. \end{alignedat} \]
1a (013)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1a}}&{}\approx{}&-0.269518956564&{}:{}&0.839961263543&{}:{}&0.429557693021&,\\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.188533109142&{}:{}&1.003040014354&{}:{}&0.185493094788&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.381418330097&{}:{}&0.696323118707&{}:{}&0.685095211389&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.088538608173&{}:{}&0.859948758263&{}:{}&0.228589849910&,\\ A^*_{\mathbf{1a}}&{}\approx{}&0.000000000000&{}:{}&1.382683432365&{}:{}&-0.382683432365&,\\B^*_{\mathbf{1a}}&{}\approx{}&-0.923879532511&{}:{}&0.000000000000&{}:{}&1.923879532511&,\\C^*_{\mathbf{1a}}&{}\approx{}&0.117316567635&{}:{}&0.882683432365&{}:{}&0.000000000000&. \end{alignedat} \]
1a (013)