Derousseau's Generalization of the Malfatti circles

Isosceles Triangle with 135° Top Angle

\(A=22.5\degree\), \(B=22.5\degree\), \(C=135\degree\).


[Top] > Isosceles Triangle with 135° Top Angle > 3 (022)

3(022)

Malfatti circles

3 (022)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.259891532474&{}:{}&0.259891532474&{}:{}&0.480216935052&, \\ P_{\mathbf{3}}&{}\approx{}&0.000093923818&{}:{}&0.998119259436&{}:{}&0.001786816746&, \\ P^-_{\mathbf{3}}&{}\approx{}&-0.007989416378&{}:{}&1.021088469546&{}:{}&-0.013099053168&, \\ P^+_{\mathbf{3}}&{}\approx{}&0.007703721666&{}:{}&0.976495643280&{}:{}&0.015800635054&, \\ Q_{\mathbf{3}}&{}\approx{}&-0.008526445804&{}:{}&0.878964600913&{}:{}&0.129561844890&, \\ I^\prime_{\mathbf{3}}&{}\approx{}&0.009089605290&{}:{}&0.937018949093&{}:{}&0.053891445616&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{3}}\)
\(P^-_{\mathbf{3}}\)
\(P^+_{\mathbf{3}}\)
\(Q_{\mathbf{3}}\)
\(I^\prime_{\mathbf{3}}\)
3 (022)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3}}&{}\approx{}&-0.531270854722&{}:{}&0.537710817439&{}:{}&0.993560037284&,\\B^\prime_{\mathbf{3}}&{}\approx{}&0.007744600231&{}:{}&0.977945244486&{}:{}&0.014310155282&,\\C^\prime_{\mathbf{3}}&{}\approx{}&2.630986313698&{}:{}&2.630986313698&{}:{}&-4.261972627396&, \\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.015443740099&{}:{}&0.982796873040&{}:{}&0.001759386861&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.005347135768&{}:{}&0.892928382147&{}:{}&0.101724482085&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.000091437238&{}:{}&0.971694615132&{}:{}&0.028213947631&, \\ A^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&0.007958327031&{}:{}&1.004933481561&{}:{}&-0.012891808592&,\\B^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&0.614466172935&{}:{}&-0.621914613235&{}:{}&1.007448440299&,\\C^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&-0.007771499506&{}:{}&0.993237573411&{}:{}&0.014533926094&, \\ A^*_{\mathbf{3}}&{}\approx{}&0.000000000000&{}:{}&0.871533517609&{}:{}&0.128466482391&,\\B^*_{\mathbf{3}}&{}\approx{}&-0.070445884989&{}:{}&0.000000000000&{}:{}&1.070445884989&,\\C^*_{\mathbf{3}}&{}\approx{}&-0.009795579104&{}:{}&1.009795579104&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3}}}{B^\prime_{\mathbf{3}}}{C^\prime_{\mathbf{3}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3}}}{B^{\prime\prime}_{\mathbf{3}}}{C^{\prime\prime}_{\mathbf{3}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3}}}{B^{\prime\prime\prime}_{\mathbf{3}}}{C^{\prime\prime\prime}_{\mathbf{3}}}\)
\(\triangle{A^*_{\mathbf{3}}}{B^*_{\mathbf{3}}}{C^*_{\mathbf{3}}}\)
3 (022)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3}}}}&{}\approx{}&2.068981672161&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3}}}}&{}\approx{}&0.029799355745&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3}}}}&{}\approx{}&10.123401438481&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.259891532474&{}:{}&0.259891532474&{}:{}&0.480216935052&,\\ A^\prime_{\mathbf{3}}&{}\approx{}&-0.531270854722&{}:{}&0.537710817439&{}:{}&0.993560037284&,\\B^\prime_{\mathbf{3}}&{}\approx{}&0.007744600231&{}:{}&0.977945244486&{}:{}&0.014310155282&,\\C^\prime_{\mathbf{3}}&{}\approx{}&2.630986313698&{}:{}&2.630986313698&{}:{}&-4.261972627396&. \end{alignedat} \]
3 (022)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3}}&{}\approx{}&0.000093923818&{}:{}&0.998119259436&{}:{}&0.001786816746&,\\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.015443740099&{}:{}&0.982796873040&{}:{}&0.001759386861&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.005347135768&{}:{}&0.892928382147&{}:{}&0.101724482085&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.000091437238&{}:{}&0.971694615132&{}:{}&0.028213947631&. \end{alignedat} \]
3 (022)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3}}&{}\approx{}&-0.007989416378&{}:{}&1.021088469546&{}:{}&-0.013099053168&,\\ A^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&0.007958327031&{}:{}&1.004933481561&{}:{}&-0.012891808592&,\\B^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&0.614466172935&{}:{}&-0.621914613235&{}:{}&1.007448440299&,\\C^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&-0.007771499506&{}:{}&0.993237573411&{}:{}&0.014533926094&. \end{alignedat} \]
3 (022)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3}}&{}\approx{}&0.007703721666&{}:{}&0.976495643280&{}:{}&0.015800635054&,\\ A^\prime_{\mathbf{3}}&{}\approx{}&-0.531270854722&{}:{}&0.537710817439&{}:{}&0.993560037284&,\\B^\prime_{\mathbf{3}}&{}\approx{}&0.007744600231&{}:{}&0.977945244486&{}:{}&0.014310155282&,\\C^\prime_{\mathbf{3}}&{}\approx{}&2.630986313698&{}:{}&2.630986313698&{}:{}&-4.261972627396&,\\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.015443740099&{}:{}&0.982796873040&{}:{}&0.001759386861&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.005347135768&{}:{}&0.892928382147&{}:{}&0.101724482085&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.000091437238&{}:{}&0.971694615132&{}:{}&0.028213947631&, \end{alignedat} \]
3 (022)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3}}&{}\approx{}&-0.008526445804&{}:{}&0.878964600913&{}:{}&0.129561844890&,\\ A^*_{\mathbf{3}}&{}\approx{}&0.000000000000&{}:{}&0.871533517609&{}:{}&0.128466482391&,\\B^*_{\mathbf{3}}&{}\approx{}&-0.070445884989&{}:{}&0.000000000000&{}:{}&1.070445884989&,\\C^*_{\mathbf{3}}&{}\approx{}&-0.009795579104&{}:{}&1.009795579104&{}:{}&0.000000000000&. \end{alignedat} \]
3 (022)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3}}&{}\approx{}&0.009089605290&{}:{}&0.937018949093&{}:{}&0.053891445616&,\\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.015443740099&{}:{}&0.982796873040&{}:{}&0.001759386861&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.005347135768&{}:{}&0.892928382147&{}:{}&0.101724482085&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.000091437238&{}:{}&0.971694615132&{}:{}&0.028213947631&,\\ A^*_{\mathbf{3}}&{}\approx{}&0.000000000000&{}:{}&0.871533517609&{}:{}&0.128466482391&,\\B^*_{\mathbf{3}}&{}\approx{}&-0.070445884989&{}:{}&0.000000000000&{}:{}&1.070445884989&,\\C^*_{\mathbf{3}}&{}\approx{}&-0.009795579104&{}:{}&1.009795579104&{}:{}&0.000000000000&. \end{alignedat} \]
3 (022)