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 > 6a (231)

6a(231)

Malfatti circles

6a (231)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.707106781187&{}:{}&0.707106781187&{}:{}&1.000000000000&, \\ P_{\mathbf{6a}}&{}\approx{}&1.018347463817&{}:{}&-0.015483994707&{}:{}&-0.002863469111&, \\ P^-_{\mathbf{6a}}&{}\approx{}&0.933038226145&{}:{}&0.020242057386&{}:{}&0.046719716469&, \\ P^+_{\mathbf{6a}}&{}\approx{}&1.113018032345&{}:{}&-0.055130412318&{}:{}&-0.057887620027&, \\ Q_{\mathbf{6a}}&{}\approx{}&1.274374312309&{}:{}&-0.379372815896&{}:{}&0.104998503587&, \\ I^\prime_{\mathbf{6a}}&{}\approx{}&1.229058233121&{}:{}&-0.151553501366&{}:{}&-0.077504731755&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{6a}}\)
\(P^-_{\mathbf{6a}}\)
\(P^+_{\mathbf{6a}}\)
\(Q_{\mathbf{6a}}\)
\(I^\prime_{\mathbf{6a}}\)
6a (231)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6a}}&{}\approx{}&1.099456183690&{}:{}&-0.041196100146&{}:{}&-0.058260083544&,\\B^\prime_{\mathbf{6a}}&{}\approx{}&6.068535592272&{}:{}&3.513669746063&{}:{}&-8.582205338335&,\\C^\prime_{\mathbf{6a}}&{}\approx{}&0.748302881333&{}:{}&-0.748302881333&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.332319281658&{}:{}&-0.280454565784&{}:{}&-0.051864715874&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.081132985117&{}:{}&-0.078092970763&{}:{}&-0.003040014354&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.132961205392&{}:{}&-0.017226699069&{}:{}&-0.115734506323&, \\ A^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.343011371888&{}:{}&0.198602885594&{}:{}&0.458385742518&,\\B^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.987553715251&{}:{}&-0.037003167890&{}:{}&0.049449452639&,\\C^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.032306156571&{}:{}&0.022395653121&{}:{}&-0.054701809692&, \\ A^*_{\mathbf{6a}}&{}\approx{}&0.000000000000&{}:{}&1.382683432365&{}:{}&-0.382683432365&,\\B^*_{\mathbf{6a}}&{}\approx{}&0.923879532511&{}:{}&0.000000000000&{}:{}&0.076120467489&,\\C^*_{\mathbf{6a}}&{}\approx{}&1.423879532511&{}:{}&-0.423879532511&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6a}}}{B^\prime_{\mathbf{6a}}}{C^\prime_{\mathbf{6a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6a}}}{B^{\prime\prime}_{\mathbf{6a}}}{C^{\prime\prime}_{\mathbf{6a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6a}}}{B^{\prime\prime\prime}_{\mathbf{6a}}}{C^{\prime\prime\prime}_{\mathbf{6a}}}\)
\(\triangle{A^*_{\mathbf{6a}}}{B^*_{\mathbf{6a}}}{C^*_{\mathbf{6a}}}\)
6a (231)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6a}}}}&{}\approx{}&-0.058260083544&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6a}}}}&{}\approx{}&-8.582205338335&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6a}}}}&{}\approx{}&-1.058260083544&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.707106781187&{}:{}&0.707106781187&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{6a}}&{}\approx{}&1.099456183690&{}:{}&-0.041196100146&{}:{}&-0.058260083544&,\\B^\prime_{\mathbf{6a}}&{}\approx{}&6.068535592272&{}:{}&3.513669746063&{}:{}&-8.582205338335&,\\C^\prime_{\mathbf{6a}}&{}\approx{}&0.748302881333&{}:{}&-0.748302881333&{}:{}&1.000000000000&. \end{alignedat} \]
6a (231)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6a}}&{}\approx{}&1.018347463817&{}:{}&-0.015483994707&{}:{}&-0.002863469111&,\\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.332319281658&{}:{}&-0.280454565784&{}:{}&-0.051864715874&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.081132985117&{}:{}&-0.078092970763&{}:{}&-0.003040014354&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.132961205392&{}:{}&-0.017226699069&{}:{}&-0.115734506323&. \end{alignedat} \]
6a (231)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6a}}&{}\approx{}&0.933038226145&{}:{}&0.020242057386&{}:{}&0.046719716469&,\\ A^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.343011371888&{}:{}&0.198602885594&{}:{}&0.458385742518&,\\B^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.987553715251&{}:{}&-0.037003167890&{}:{}&0.049449452639&,\\C^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.032306156571&{}:{}&0.022395653121&{}:{}&-0.054701809692&. \end{alignedat} \]
6a (231)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6a}}&{}\approx{}&1.113018032345&{}:{}&-0.055130412318&{}:{}&-0.057887620027&,\\ A^\prime_{\mathbf{6a}}&{}\approx{}&1.099456183690&{}:{}&-0.041196100146&{}:{}&-0.058260083544&,\\B^\prime_{\mathbf{6a}}&{}\approx{}&6.068535592272&{}:{}&3.513669746063&{}:{}&-8.582205338335&,\\C^\prime_{\mathbf{6a}}&{}\approx{}&0.748302881333&{}:{}&-0.748302881333&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.332319281658&{}:{}&-0.280454565784&{}:{}&-0.051864715874&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.081132985117&{}:{}&-0.078092970763&{}:{}&-0.003040014354&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.132961205392&{}:{}&-0.017226699069&{}:{}&-0.115734506323&, \end{alignedat} \]
6a (231)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6a}}&{}\approx{}&1.274374312309&{}:{}&-0.379372815896&{}:{}&0.104998503587&,\\ A^*_{\mathbf{6a}}&{}\approx{}&0.000000000000&{}:{}&1.382683432365&{}:{}&-0.382683432365&,\\B^*_{\mathbf{6a}}&{}\approx{}&0.923879532511&{}:{}&0.000000000000&{}:{}&0.076120467489&,\\C^*_{\mathbf{6a}}&{}\approx{}&1.423879532511&{}:{}&-0.423879532511&{}:{}&0.000000000000&. \end{alignedat} \]
6a (231)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6a}}&{}\approx{}&1.229058233121&{}:{}&-0.151553501366&{}:{}&-0.077504731755&,\\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.332319281658&{}:{}&-0.280454565784&{}:{}&-0.051864715874&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.081132985117&{}:{}&-0.078092970763&{}:{}&-0.003040014354&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.132961205392&{}:{}&-0.017226699069&{}:{}&-0.115734506323&,\\ A^*_{\mathbf{6a}}&{}\approx{}&0.000000000000&{}:{}&1.382683432365&{}:{}&-0.382683432365&,\\B^*_{\mathbf{6a}}&{}\approx{}&0.923879532511&{}:{}&0.000000000000&{}:{}&0.076120467489&,\\C^*_{\mathbf{6a}}&{}\approx{}&1.423879532511&{}:{}&-0.423879532511&{}:{}&0.000000000000&. \end{alignedat} \]
6a (231)