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 > 0b (101)

0b(101)

Malfatti circles

0b (101)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.707106781187&{}:{}&-0.707106781187&{}:{}&1.000000000000&, \\ P_{\mathbf{b}}&{}\approx{}&0.708594513162&{}:{}&-0.366005451906&{}:{}&0.657410938744&, \\ P^-_{\mathbf{b}}&{}\approx{}&0.708992238998&{}:{}&-0.274816438215&{}:{}&0.565824199217&, \\ P^+_{\mathbf{b}}&{}\approx{}&0.708335353350&{}:{}&-0.425424593061&{}:{}&0.717089239711&, \\ Q_{\mathbf{b}}&{}\approx{}&0.756303214887&{}:{}&-0.225146471905&{}:{}&0.468843257019&, \\ I^\prime_{\mathbf{b}}&{}\approx{}&0.700890186899&{}:{}&-0.503726575887&{}:{}&0.802836388989&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{b}}\)
\(P^-_{\mathbf{b}}\)
\(P^+_{\mathbf{b}}\)
\(Q_{\mathbf{b}}\)
\(I^\prime_{\mathbf{b}}\)
0b (101)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{b}}&{}\approx{}&0.900543816310&{}:{}&-0.240108467526&{}:{}&0.339564651216&,\\B^\prime_{\mathbf{b}}&{}\approx{}&1.041196100146&{}:{}&-1.513669746063&{}:{}&1.472473645917&,\\C^\prime_{\mathbf{b}}&{}\approx{}&0.334089318960&{}:{}&-0.334089318960&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.505862261271&{}:{}&-0.620637271897&{}:{}&1.114775010625&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.663673308476&{}:{}&-0.279407814799&{}:{}&0.615734506323&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.926901175191&{}:{}&-0.478765891065&{}:{}&0.551864715874&, \\ A^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.393788873157&{}:{}&-0.572482266837&{}:{}&1.178693393680&,\\B^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.652979274079&{}:{}&-0.174101304107&{}:{}&0.521122030028&,\\C^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&1.010661837698&{}:{}&-0.391748274803&{}:{}&0.381086437105&, \\ A^*_{\mathbf{b}}&{}\approx{}&0.000000000000&{}:{}&-0.923879532511&{}:{}&1.923879532511&,\\B^*_{\mathbf{b}}&{}\approx{}&0.617316567635&{}:{}&0.000000000000&{}:{}&0.382683432365&,\\C^*_{\mathbf{b}}&{}\approx{}&1.423879532511&{}:{}&-0.423879532511&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{b}}}{B^\prime_{\mathbf{b}}}{C^\prime_{\mathbf{b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{b}}}{B^{\prime\prime}_{\mathbf{b}}}{C^{\prime\prime}_{\mathbf{b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{b}}}{B^{\prime\prime\prime}_{\mathbf{b}}}{C^{\prime\prime\prime}_{\mathbf{b}}}\)
\(\triangle{A^*_{\mathbf{b}}}{B^*_{\mathbf{b}}}{C^*_{\mathbf{b}}}\)
0b (101)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{b}}}}&{}\approx{}&0.339564651216&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{b}}}}&{}\approx{}&1.472473645917&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{b}}}}&{}\approx{}&0.472473645917&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.707106781187&{}:{}&-0.707106781187&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{b}}&{}\approx{}&0.900543816310&{}:{}&-0.240108467526&{}:{}&0.339564651216&,\\B^\prime_{\mathbf{b}}&{}\approx{}&1.041196100146&{}:{}&-1.513669746063&{}:{}&1.472473645917&,\\C^\prime_{\mathbf{b}}&{}\approx{}&0.334089318960&{}:{}&-0.334089318960&{}:{}&1.000000000000&. \end{alignedat} \]
0b (101)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{b}}&{}\approx{}&0.708594513162&{}:{}&-0.366005451906&{}:{}&0.657410938744&,\\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.505862261271&{}:{}&-0.620637271897&{}:{}&1.114775010625&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.663673308476&{}:{}&-0.279407814799&{}:{}&0.615734506323&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.926901175191&{}:{}&-0.478765891065&{}:{}&0.551864715874&. \end{alignedat} \]
0b (101)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{b}}&{}\approx{}&0.708992238998&{}:{}&-0.274816438215&{}:{}&0.565824199217&,\\ A^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.393788873157&{}:{}&-0.572482266837&{}:{}&1.178693393680&,\\B^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.652979274079&{}:{}&-0.174101304107&{}:{}&0.521122030028&,\\C^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&1.010661837698&{}:{}&-0.391748274803&{}:{}&0.381086437105&. \end{alignedat} \]
0b (101)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{b}}&{}\approx{}&0.708335353350&{}:{}&-0.425424593061&{}:{}&0.717089239711&,\\ A^\prime_{\mathbf{b}}&{}\approx{}&0.900543816310&{}:{}&-0.240108467526&{}:{}&0.339564651216&,\\B^\prime_{\mathbf{b}}&{}\approx{}&1.041196100146&{}:{}&-1.513669746063&{}:{}&1.472473645917&,\\C^\prime_{\mathbf{b}}&{}\approx{}&0.334089318960&{}:{}&-0.334089318960&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.505862261271&{}:{}&-0.620637271897&{}:{}&1.114775010625&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.663673308476&{}:{}&-0.279407814799&{}:{}&0.615734506323&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.926901175191&{}:{}&-0.478765891065&{}:{}&0.551864715874&, \end{alignedat} \]
0b (101)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{b}}&{}\approx{}&0.756303214887&{}:{}&-0.225146471905&{}:{}&0.468843257019&,\\ A^*_{\mathbf{b}}&{}\approx{}&0.000000000000&{}:{}&-0.923879532511&{}:{}&1.923879532511&,\\B^*_{\mathbf{b}}&{}\approx{}&0.617316567635&{}:{}&0.000000000000&{}:{}&0.382683432365&,\\C^*_{\mathbf{b}}&{}\approx{}&1.423879532511&{}:{}&-0.423879532511&{}:{}&0.000000000000&. \end{alignedat} \]
0b (101)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{b}}&{}\approx{}&0.700890186899&{}:{}&-0.503726575887&{}:{}&0.802836388989&,\\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.505862261271&{}:{}&-0.620637271897&{}:{}&1.114775010625&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.663673308476&{}:{}&-0.279407814799&{}:{}&0.615734506323&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.926901175191&{}:{}&-0.478765891065&{}:{}&0.551864715874&,\\ A^*_{\mathbf{b}}&{}\approx{}&0.000000000000&{}:{}&-0.923879532511&{}:{}&1.923879532511&,\\B^*_{\mathbf{b}}&{}\approx{}&0.617316567635&{}:{}&0.000000000000&{}:{}&0.382683432365&,\\C^*_{\mathbf{b}}&{}\approx{}&1.423879532511&{}:{}&-0.423879532511&{}:{}&0.000000000000&. \end{alignedat} \]
0b (101)