Derousseau's Generalization of the Malfatti circles

Isosceles Triangle with 135° Top Angle

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


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5(202)

Malfatti circles

5 (202)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.259891532474&{}:{}&0.259891532474&{}:{}&0.480216935052&, \\ P_{\mathbf{5}}&{}\approx{}&0.998119259436&{}:{}&0.000093923818&{}:{}&0.001786816746&, \\ P^-_{\mathbf{5}}&{}\approx{}&1.021088469546&{}:{}&-0.007989416378&{}:{}&-0.013099053168&, \\ P^+_{\mathbf{5}}&{}\approx{}&0.976495643280&{}:{}&0.007703721666&{}:{}&0.015800635054&, \\ Q_{\mathbf{5}}&{}\approx{}&0.878964600913&{}:{}&-0.008526445804&{}:{}&0.129561844890&, \\ I^\prime_{\mathbf{5}}&{}\approx{}&0.937018949093&{}:{}&0.009089605290&{}:{}&0.053891445616&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{5}}\)
\(P^-_{\mathbf{5}}\)
\(P^+_{\mathbf{5}}\)
\(Q_{\mathbf{5}}\)
\(I^\prime_{\mathbf{5}}\)
5 (202)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5}}&{}\approx{}&0.977945244486&{}:{}&0.007744600231&{}:{}&0.014310155282&,\\B^\prime_{\mathbf{5}}&{}\approx{}&0.537710817439&{}:{}&-0.531270854722&{}:{}&0.993560037284&,\\C^\prime_{\mathbf{5}}&{}\approx{}&2.630986313698&{}:{}&2.630986313698&{}:{}&-4.261972627396&, \\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.892928382147&{}:{}&0.005347135768&{}:{}&0.101724482085&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.982796873040&{}:{}&0.015443740099&{}:{}&0.001759386861&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.971694615132&{}:{}&0.000091437238&{}:{}&0.028213947631&, \\ A^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&-0.621914613235&{}:{}&0.614466172935&{}:{}&1.007448440299&,\\B^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&1.004933481561&{}:{}&0.007958327031&{}:{}&-0.012891808592&,\\C^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&0.993237573411&{}:{}&-0.007771499506&{}:{}&0.014533926094&, \\ A^*_{\mathbf{5}}&{}\approx{}&0.000000000000&{}:{}&-0.070445884989&{}:{}&1.070445884989&,\\B^*_{\mathbf{5}}&{}\approx{}&0.871533517609&{}:{}&0.000000000000&{}:{}&0.128466482391&,\\C^*_{\mathbf{5}}&{}\approx{}&1.009795579104&{}:{}&-0.009795579104&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5}}}{B^\prime_{\mathbf{5}}}{C^\prime_{\mathbf{5}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5}}}{B^{\prime\prime}_{\mathbf{5}}}{C^{\prime\prime}_{\mathbf{5}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5}}}{B^{\prime\prime\prime}_{\mathbf{5}}}{C^{\prime\prime\prime}_{\mathbf{5}}}\)
\(\triangle{A^*_{\mathbf{5}}}{B^*_{\mathbf{5}}}{C^*_{\mathbf{5}}}\)
5 (202)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5}}}}&{}\approx{}&0.029799355745&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5}}}}&{}\approx{}&2.068981672161&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5}}}}&{}\approx{}&10.123401438481&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.259891532474&{}:{}&0.259891532474&{}:{}&0.480216935052&,\\ A^\prime_{\mathbf{5}}&{}\approx{}&0.977945244486&{}:{}&0.007744600231&{}:{}&0.014310155282&,\\B^\prime_{\mathbf{5}}&{}\approx{}&0.537710817439&{}:{}&-0.531270854722&{}:{}&0.993560037284&,\\C^\prime_{\mathbf{5}}&{}\approx{}&2.630986313698&{}:{}&2.630986313698&{}:{}&-4.261972627396&. \end{alignedat} \]
5 (202)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5}}&{}\approx{}&0.998119259436&{}:{}&0.000093923818&{}:{}&0.001786816746&,\\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.892928382147&{}:{}&0.005347135768&{}:{}&0.101724482085&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.982796873040&{}:{}&0.015443740099&{}:{}&0.001759386861&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.971694615132&{}:{}&0.000091437238&{}:{}&0.028213947631&. \end{alignedat} \]
5 (202)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5}}&{}\approx{}&1.021088469546&{}:{}&-0.007989416378&{}:{}&-0.013099053168&,\\ A^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&-0.621914613235&{}:{}&0.614466172935&{}:{}&1.007448440299&,\\B^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&1.004933481561&{}:{}&0.007958327031&{}:{}&-0.012891808592&,\\C^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&0.993237573411&{}:{}&-0.007771499506&{}:{}&0.014533926094&. \end{alignedat} \]
5 (202)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5}}&{}\approx{}&0.976495643280&{}:{}&0.007703721666&{}:{}&0.015800635054&,\\ A^\prime_{\mathbf{5}}&{}\approx{}&0.977945244486&{}:{}&0.007744600231&{}:{}&0.014310155282&,\\B^\prime_{\mathbf{5}}&{}\approx{}&0.537710817439&{}:{}&-0.531270854722&{}:{}&0.993560037284&,\\C^\prime_{\mathbf{5}}&{}\approx{}&2.630986313698&{}:{}&2.630986313698&{}:{}&-4.261972627396&,\\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.892928382147&{}:{}&0.005347135768&{}:{}&0.101724482085&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.982796873040&{}:{}&0.015443740099&{}:{}&0.001759386861&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.971694615132&{}:{}&0.000091437238&{}:{}&0.028213947631&, \end{alignedat} \]
5 (202)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5}}&{}\approx{}&0.878964600913&{}:{}&-0.008526445804&{}:{}&0.129561844890&,\\ A^*_{\mathbf{5}}&{}\approx{}&0.000000000000&{}:{}&-0.070445884989&{}:{}&1.070445884989&,\\B^*_{\mathbf{5}}&{}\approx{}&0.871533517609&{}:{}&0.000000000000&{}:{}&0.128466482391&,\\C^*_{\mathbf{5}}&{}\approx{}&1.009795579104&{}:{}&-0.009795579104&{}:{}&0.000000000000&. \end{alignedat} \]
5 (202)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5}}&{}\approx{}&0.937018949093&{}:{}&0.009089605290&{}:{}&0.053891445616&,\\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.892928382147&{}:{}&0.005347135768&{}:{}&0.101724482085&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.982796873040&{}:{}&0.015443740099&{}:{}&0.001759386861&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.971694615132&{}:{}&0.000091437238&{}:{}&0.028213947631&,\\ A^*_{\mathbf{5}}&{}\approx{}&0.000000000000&{}:{}&-0.070445884989&{}:{}&1.070445884989&,\\B^*_{\mathbf{5}}&{}\approx{}&0.871533517609&{}:{}&0.000000000000&{}:{}&0.128466482391&,\\C^*_{\mathbf{5}}&{}\approx{}&1.009795579104&{}:{}&-0.009795579104&{}:{}&0.000000000000&. \end{alignedat} \]
5 (202)