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 > 6b (321)

6b(321)

Malfatti circles

6b (321)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.707106781187&{}:{}&-0.707106781187&{}:{}&1.000000000000&, \\ P_{\mathbf{6b}}&{}\approx{}&-0.015483994707&{}:{}&1.018347463817&{}:{}&-0.002863469111&, \\ P^-_{\mathbf{6b}}&{}\approx{}&0.020242057386&{}:{}&0.933038226145&{}:{}&0.046719716469&, \\ P^+_{\mathbf{6b}}&{}\approx{}&-0.055130412318&{}:{}&1.113018032345&{}:{}&-0.057887620027&, \\ Q_{\mathbf{6b}}&{}\approx{}&-0.379372815896&{}:{}&1.274374312309&{}:{}&0.104998503587&, \\ I^\prime_{\mathbf{6b}}&{}\approx{}&-0.151553501366&{}:{}&1.229058233121&{}:{}&-0.077504731755&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{6b}}\)
\(P^-_{\mathbf{6b}}\)
\(P^+_{\mathbf{6b}}\)
\(Q_{\mathbf{6b}}\)
\(I^\prime_{\mathbf{6b}}\)
6b (321)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6b}}&{}\approx{}&3.513669746063&{}:{}&6.068535592272&{}:{}&-8.582205338335&,\\B^\prime_{\mathbf{6b}}&{}\approx{}&-0.041196100146&{}:{}&1.099456183690&{}:{}&-0.058260083544&,\\C^\prime_{\mathbf{6b}}&{}\approx{}&-0.748302881333&{}:{}&0.748302881333&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.078092970763&{}:{}&1.081132985117&{}:{}&-0.003040014354&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.280454565784&{}:{}&1.332319281658&{}:{}&-0.051864715874&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.017226699069&{}:{}&1.132961205392&{}:{}&-0.115734506323&, \\ A^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.037003167890&{}:{}&0.987553715251&{}:{}&0.049449452639&,\\B^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.198602885594&{}:{}&0.343011371888&{}:{}&0.458385742518&,\\C^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.022395653121&{}:{}&1.032306156571&{}:{}&-0.054701809692&, \\ A^*_{\mathbf{6b}}&{}\approx{}&0.000000000000&{}:{}&0.923879532511&{}:{}&0.076120467489&,\\B^*_{\mathbf{6b}}&{}\approx{}&1.382683432365&{}:{}&0.000000000000&{}:{}&-0.382683432365&,\\C^*_{\mathbf{6b}}&{}\approx{}&-0.423879532511&{}:{}&1.423879532511&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6b}}}{B^\prime_{\mathbf{6b}}}{C^\prime_{\mathbf{6b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6b}}}{B^{\prime\prime}_{\mathbf{6b}}}{C^{\prime\prime}_{\mathbf{6b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6b}}}{B^{\prime\prime\prime}_{\mathbf{6b}}}{C^{\prime\prime\prime}_{\mathbf{6b}}}\)
\(\triangle{A^*_{\mathbf{6b}}}{B^*_{\mathbf{6b}}}{C^*_{\mathbf{6b}}}\)
6b (321)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6b}}}}&{}\approx{}&-8.582205338335&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6b}}}}&{}\approx{}&-0.058260083544&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6b}}}}&{}\approx{}&-1.058260083544&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.707106781187&{}:{}&-0.707106781187&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{6b}}&{}\approx{}&3.513669746063&{}:{}&6.068535592272&{}:{}&-8.582205338335&,\\B^\prime_{\mathbf{6b}}&{}\approx{}&-0.041196100146&{}:{}&1.099456183690&{}:{}&-0.058260083544&,\\C^\prime_{\mathbf{6b}}&{}\approx{}&-0.748302881333&{}:{}&0.748302881333&{}:{}&1.000000000000&. \end{alignedat} \]
6b (321)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6b}}&{}\approx{}&-0.015483994707&{}:{}&1.018347463817&{}:{}&-0.002863469111&,\\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.078092970763&{}:{}&1.081132985117&{}:{}&-0.003040014354&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.280454565784&{}:{}&1.332319281658&{}:{}&-0.051864715874&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.017226699069&{}:{}&1.132961205392&{}:{}&-0.115734506323&. \end{alignedat} \]
6b (321)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6b}}&{}\approx{}&0.020242057386&{}:{}&0.933038226145&{}:{}&0.046719716469&,\\ A^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.037003167890&{}:{}&0.987553715251&{}:{}&0.049449452639&,\\B^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.198602885594&{}:{}&0.343011371888&{}:{}&0.458385742518&,\\C^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.022395653121&{}:{}&1.032306156571&{}:{}&-0.054701809692&. \end{alignedat} \]
6b (321)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6b}}&{}\approx{}&-0.055130412318&{}:{}&1.113018032345&{}:{}&-0.057887620027&,\\ A^\prime_{\mathbf{6b}}&{}\approx{}&3.513669746063&{}:{}&6.068535592272&{}:{}&-8.582205338335&,\\B^\prime_{\mathbf{6b}}&{}\approx{}&-0.041196100146&{}:{}&1.099456183690&{}:{}&-0.058260083544&,\\C^\prime_{\mathbf{6b}}&{}\approx{}&-0.748302881333&{}:{}&0.748302881333&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.078092970763&{}:{}&1.081132985117&{}:{}&-0.003040014354&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.280454565784&{}:{}&1.332319281658&{}:{}&-0.051864715874&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.017226699069&{}:{}&1.132961205392&{}:{}&-0.115734506323&, \end{alignedat} \]
6b (321)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6b}}&{}\approx{}&-0.379372815896&{}:{}&1.274374312309&{}:{}&0.104998503587&,\\ A^*_{\mathbf{6b}}&{}\approx{}&0.000000000000&{}:{}&0.923879532511&{}:{}&0.076120467489&,\\B^*_{\mathbf{6b}}&{}\approx{}&1.382683432365&{}:{}&0.000000000000&{}:{}&-0.382683432365&,\\C^*_{\mathbf{6b}}&{}\approx{}&-0.423879532511&{}:{}&1.423879532511&{}:{}&0.000000000000&. \end{alignedat} \]
6b (321)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6b}}&{}\approx{}&-0.151553501366&{}:{}&1.229058233121&{}:{}&-0.077504731755&,\\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.078092970763&{}:{}&1.081132985117&{}:{}&-0.003040014354&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.280454565784&{}:{}&1.332319281658&{}:{}&-0.051864715874&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.017226699069&{}:{}&1.132961205392&{}:{}&-0.115734506323&,\\ A^*_{\mathbf{6b}}&{}\approx{}&0.000000000000&{}:{}&0.923879532511&{}:{}&0.076120467489&,\\B^*_{\mathbf{6b}}&{}\approx{}&1.382683432365&{}:{}&0.000000000000&{}:{}&-0.382683432365&,\\C^*_{\mathbf{6b}}&{}\approx{}&-0.423879532511&{}:{}&1.423879532511&{}:{}&0.000000000000&. \end{alignedat} \]
6b (321)