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 > 1b (103)

1b(103)

Malfatti circles

1b (103)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.707106781187&{}:{}&-0.707106781187&{}:{}&1.000000000000&, \\ P_{\mathbf{1b}}&{}\approx{}&0.924237568416&{}:{}&-0.095157655782&{}:{}&0.170920087366&, \\ P^-_{\mathbf{1b}}&{}\approx{}&0.954645041042&{}:{}&-0.009458955395&{}:{}&0.054813914352&, \\ P^+_{\mathbf{1b}}&{}\approx{}&0.900483296637&{}:{}&-0.162105351339&{}:{}&0.261622054702&, \\ Q_{\mathbf{1b}}&{}\approx{}&1.168032800651&{}:{}&0.155242065312&{}:{}&-0.323274865964&, \\ I^\prime_{\mathbf{1b}}&{}\approx{}&0.839961263543&{}:{}&-0.269518956564&{}:{}&0.429557693021&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{1b}}\)
\(P^-_{\mathbf{1b}}\)
\(P^+_{\mathbf{1b}}\)
\(Q_{\mathbf{1b}}\)
\(I^\prime_{\mathbf{1b}}\)
1b (103)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1b}}&{}\approx{}&0.958803899854&{}:{}&-0.099456183690&{}:{}&0.140652283836&,\\B^\prime_{\mathbf{1b}}&{}\approx{}&0.431277545771&{}:{}&-0.041196100146&{}:{}&0.609918554376&,\\C^\prime_{\mathbf{1b}}&{}\approx{}&1.806562964876&{}:{}&-1.806562964876&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.696323118707&{}:{}&-0.381418330097&{}:{}&0.685095211389&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.003040014354&{}:{}&-0.188533109142&{}:{}&0.185493094788&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.859948758263&{}:{}&-0.088538608173&{}:{}&0.228589849910&, \\ A^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.685863425930&{}:{}&-0.065514420258&{}:{}&0.379650994327&,\\B^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.048560208790&{}:{}&-0.108766554612&{}:{}&0.060206345821&,\\C^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.879672098223&{}:{}&-0.008716097378&{}:{}&0.129043999154&, \\ A^*_{\mathbf{1b}}&{}\approx{}&0.000000000000&{}:{}&-0.923879532511&{}:{}&1.923879532511&,\\B^*_{\mathbf{1b}}&{}\approx{}&1.382683432365&{}:{}&0.000000000000&{}:{}&-0.382683432365&,\\C^*_{\mathbf{1b}}&{}\approx{}&0.882683432365&{}:{}&0.117316567635&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1b}}}{B^\prime_{\mathbf{1b}}}{C^\prime_{\mathbf{1b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1b}}}{B^{\prime\prime}_{\mathbf{1b}}}{C^{\prime\prime}_{\mathbf{1b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1b}}}{B^{\prime\prime\prime}_{\mathbf{1b}}}{C^{\prime\prime\prime}_{\mathbf{1b}}}\)
\(\triangle{A^*_{\mathbf{1b}}}{B^*_{\mathbf{1b}}}{C^*_{\mathbf{1b}}}\)
1b (103)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1b}}}}&{}\approx{}&0.140652283836&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1b}}}}&{}\approx{}&0.609918554376&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1b}}}}&{}\approx{}&2.554865846209&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.707106781187&{}:{}&-0.707106781187&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{1b}}&{}\approx{}&0.958803899854&{}:{}&-0.099456183690&{}:{}&0.140652283836&,\\B^\prime_{\mathbf{1b}}&{}\approx{}&0.431277545771&{}:{}&-0.041196100146&{}:{}&0.609918554376&,\\C^\prime_{\mathbf{1b}}&{}\approx{}&1.806562964876&{}:{}&-1.806562964876&{}:{}&1.000000000000&. \end{alignedat} \]
1b (103)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1b}}&{}\approx{}&0.924237568416&{}:{}&-0.095157655782&{}:{}&0.170920087366&,\\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.696323118707&{}:{}&-0.381418330097&{}:{}&0.685095211389&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.003040014354&{}:{}&-0.188533109142&{}:{}&0.185493094788&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.859948758263&{}:{}&-0.088538608173&{}:{}&0.228589849910&. \end{alignedat} \]
1b (103)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1b}}&{}\approx{}&0.954645041042&{}:{}&-0.009458955395&{}:{}&0.054813914352&,\\ A^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.685863425930&{}:{}&-0.065514420258&{}:{}&0.379650994327&,\\B^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.048560208790&{}:{}&-0.108766554612&{}:{}&0.060206345821&,\\C^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.879672098223&{}:{}&-0.008716097378&{}:{}&0.129043999154&. \end{alignedat} \]
1b (103)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1b}}&{}\approx{}&0.900483296637&{}:{}&-0.162105351339&{}:{}&0.261622054702&,\\ A^\prime_{\mathbf{1b}}&{}\approx{}&0.958803899854&{}:{}&-0.099456183690&{}:{}&0.140652283836&,\\B^\prime_{\mathbf{1b}}&{}\approx{}&0.431277545771&{}:{}&-0.041196100146&{}:{}&0.609918554376&,\\C^\prime_{\mathbf{1b}}&{}\approx{}&1.806562964876&{}:{}&-1.806562964876&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.696323118707&{}:{}&-0.381418330097&{}:{}&0.685095211389&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.003040014354&{}:{}&-0.188533109142&{}:{}&0.185493094788&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.859948758263&{}:{}&-0.088538608173&{}:{}&0.228589849910&, \end{alignedat} \]
1b (103)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1b}}&{}\approx{}&1.168032800651&{}:{}&0.155242065312&{}:{}&-0.323274865964&,\\ A^*_{\mathbf{1b}}&{}\approx{}&0.000000000000&{}:{}&-0.923879532511&{}:{}&1.923879532511&,\\B^*_{\mathbf{1b}}&{}\approx{}&1.382683432365&{}:{}&0.000000000000&{}:{}&-0.382683432365&,\\C^*_{\mathbf{1b}}&{}\approx{}&0.882683432365&{}:{}&0.117316567635&{}:{}&0.000000000000&. \end{alignedat} \]
1b (103)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1b}}&{}\approx{}&0.839961263543&{}:{}&-0.269518956564&{}:{}&0.429557693021&,\\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.696323118707&{}:{}&-0.381418330097&{}:{}&0.685095211389&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.003040014354&{}:{}&-0.188533109142&{}:{}&0.185493094788&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.859948758263&{}:{}&-0.088538608173&{}:{}&0.228589849910&,\\ A^*_{\mathbf{1b}}&{}\approx{}&0.000000000000&{}:{}&-0.923879532511&{}:{}&1.923879532511&,\\B^*_{\mathbf{1b}}&{}\approx{}&1.382683432365&{}:{}&0.000000000000&{}:{}&-0.382683432365&,\\C^*_{\mathbf{1b}}&{}\approx{}&0.882683432365&{}:{}&0.117316567635&{}:{}&0.000000000000&. \end{alignedat} \]
1b (103)