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 > 0a (011)

0a(011)

Malfatti circles

0a (011)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.707106781187&{}:{}&0.707106781187&{}:{}&1.000000000000&, \\ P_{\mathbf{a}}&{}\approx{}&-0.366005451906&{}:{}&0.708594513162&{}:{}&0.657410938744&, \\ P^-_{\mathbf{a}}&{}\approx{}&-0.274816438215&{}:{}&0.708992238998&{}:{}&0.565824199217&, \\ P^+_{\mathbf{a}}&{}\approx{}&-0.425424593061&{}:{}&0.708335353350&{}:{}&0.717089239711&, \\ Q_{\mathbf{a}}&{}\approx{}&-0.225146471905&{}:{}&0.756303214887&{}:{}&0.468843257019&, \\ I^\prime_{\mathbf{a}}&{}\approx{}&-0.503726575887&{}:{}&0.700890186899&{}:{}&0.802836388989&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{a}}\)
\(P^-_{\mathbf{a}}\)
\(P^+_{\mathbf{a}}\)
\(Q_{\mathbf{a}}\)
\(I^\prime_{\mathbf{a}}\)
0a (011)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{a}}&{}\approx{}&-1.513669746063&{}:{}&1.041196100146&{}:{}&1.472473645917&,\\B^\prime_{\mathbf{a}}&{}\approx{}&-0.240108467526&{}:{}&0.900543816310&{}:{}&0.339564651216&,\\C^\prime_{\mathbf{a}}&{}\approx{}&-0.334089318960&{}:{}&0.334089318960&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.279407814799&{}:{}&0.663673308476&{}:{}&0.615734506323&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.620637271897&{}:{}&0.505862261271&{}:{}&1.114775010625&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.478765891065&{}:{}&0.926901175191&{}:{}&0.551864715874&, \\ A^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.174101304107&{}:{}&0.652979274079&{}:{}&0.521122030028&,\\B^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.572482266837&{}:{}&0.393788873157&{}:{}&1.178693393680&,\\C^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.391748274803&{}:{}&1.010661837698&{}:{}&0.381086437105&, \\ A^*_{\mathbf{a}}&{}\approx{}&0.000000000000&{}:{}&0.617316567635&{}:{}&0.382683432365&,\\B^*_{\mathbf{a}}&{}\approx{}&-0.923879532511&{}:{}&0.000000000000&{}:{}&1.923879532511&,\\C^*_{\mathbf{a}}&{}\approx{}&-0.423879532511&{}:{}&1.423879532511&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{a}}}{B^\prime_{\mathbf{a}}}{C^\prime_{\mathbf{a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{a}}}{B^{\prime\prime}_{\mathbf{a}}}{C^{\prime\prime}_{\mathbf{a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{a}}}{B^{\prime\prime\prime}_{\mathbf{a}}}{C^{\prime\prime\prime}_{\mathbf{a}}}\)
\(\triangle{A^*_{\mathbf{a}}}{B^*_{\mathbf{a}}}{C^*_{\mathbf{a}}}\)
0a (011)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{a}}}}&{}\approx{}&1.472473645917&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{a}}}}&{}\approx{}&0.339564651216&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{a}}}}&{}\approx{}&0.472473645917&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.707106781187&{}:{}&0.707106781187&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{a}}&{}\approx{}&-1.513669746063&{}:{}&1.041196100146&{}:{}&1.472473645917&,\\B^\prime_{\mathbf{a}}&{}\approx{}&-0.240108467526&{}:{}&0.900543816310&{}:{}&0.339564651216&,\\C^\prime_{\mathbf{a}}&{}\approx{}&-0.334089318960&{}:{}&0.334089318960&{}:{}&1.000000000000&. \end{alignedat} \]
0a (011)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{a}}&{}\approx{}&-0.366005451906&{}:{}&0.708594513162&{}:{}&0.657410938744&,\\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.279407814799&{}:{}&0.663673308476&{}:{}&0.615734506323&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.620637271897&{}:{}&0.505862261271&{}:{}&1.114775010625&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.478765891065&{}:{}&0.926901175191&{}:{}&0.551864715874&. \end{alignedat} \]
0a (011)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{a}}&{}\approx{}&-0.274816438215&{}:{}&0.708992238998&{}:{}&0.565824199217&,\\ A^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.174101304107&{}:{}&0.652979274079&{}:{}&0.521122030028&,\\B^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.572482266837&{}:{}&0.393788873157&{}:{}&1.178693393680&,\\C^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.391748274803&{}:{}&1.010661837698&{}:{}&0.381086437105&. \end{alignedat} \]
0a (011)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{a}}&{}\approx{}&-0.425424593061&{}:{}&0.708335353350&{}:{}&0.717089239711&,\\ A^\prime_{\mathbf{a}}&{}\approx{}&-1.513669746063&{}:{}&1.041196100146&{}:{}&1.472473645917&,\\B^\prime_{\mathbf{a}}&{}\approx{}&-0.240108467526&{}:{}&0.900543816310&{}:{}&0.339564651216&,\\C^\prime_{\mathbf{a}}&{}\approx{}&-0.334089318960&{}:{}&0.334089318960&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.279407814799&{}:{}&0.663673308476&{}:{}&0.615734506323&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.620637271897&{}:{}&0.505862261271&{}:{}&1.114775010625&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.478765891065&{}:{}&0.926901175191&{}:{}&0.551864715874&, \end{alignedat} \]
0a (011)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{a}}&{}\approx{}&-0.225146471905&{}:{}&0.756303214887&{}:{}&0.468843257019&,\\ A^*_{\mathbf{a}}&{}\approx{}&0.000000000000&{}:{}&0.617316567635&{}:{}&0.382683432365&,\\B^*_{\mathbf{a}}&{}\approx{}&-0.923879532511&{}:{}&0.000000000000&{}:{}&1.923879532511&,\\C^*_{\mathbf{a}}&{}\approx{}&-0.423879532511&{}:{}&1.423879532511&{}:{}&0.000000000000&. \end{alignedat} \]
0a (011)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{a}}&{}\approx{}&-0.503726575887&{}:{}&0.700890186899&{}:{}&0.802836388989&,\\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.279407814799&{}:{}&0.663673308476&{}:{}&0.615734506323&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.620637271897&{}:{}&0.505862261271&{}:{}&1.114775010625&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.478765891065&{}:{}&0.926901175191&{}:{}&0.551864715874&,\\ A^*_{\mathbf{a}}&{}\approx{}&0.000000000000&{}:{}&0.617316567635&{}:{}&0.382683432365&,\\B^*_{\mathbf{a}}&{}\approx{}&-0.923879532511&{}:{}&0.000000000000&{}:{}&1.923879532511&,\\C^*_{\mathbf{a}}&{}\approx{}&-0.423879532511&{}:{}&1.423879532511&{}:{}&0.000000000000&. \end{alignedat} \]
0a (011)