Derousseau's Generalization of the Malfatti circles

Half of an Equilateral Triangle

\(A=30\degree\), \(B=60\degree\), \(C=90\degree\).


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\(\mathbf{5b}\) \((303)\)

Malfatti circles

5b (303)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.788675134595&{}:{}&-1.366025403784&{}:{}&1.577350269190&, \\ P_{\mathbf{5b}}&{}\approx{}&0.073887393036&{}:{}&-0.020190374287&{}:{}&0.946302981250&, \\ P^-_{\mathbf{5b}}&{}\approx{}&0.025400970204&{}:{}&0.071102078716&{}:{}&0.903496951080&, \\ P^+_{\mathbf{5b}}&{}\approx{}&0.116581632223&{}:{}&-0.100577039186&{}:{}&0.983995406963&, \\ Q_{\mathbf{5b}}&{}\approx{}&0.327000866942&{}:{}&0.067232897738&{}:{}&0.605766235320&, \\ I^\prime_{\mathbf{5b}}&{}\approx{}&0.186111137025&{}:{}&-0.128038238075&{}:{}&0.941927101051&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{5b}}\)
\(P^-_{\mathbf{5b}}\)
\(P^+_{\mathbf{5b}}\)
\(Q_{\mathbf{5b}}\)
\(I^\prime_{\mathbf{5b}}\)
5b (303)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5b}}&{}\approx{}&0.639245889233&{}:{}&-2.331951230074&{}:{}&2.692705340840&,\\B^\prime_{\mathbf{5b}}&{}\approx{}&0.230997598770&{}:{}&0.307007203691&{}:{}&0.461995197539&,\\C^\prime_{\mathbf{5b}}&{}\approx{}&0.061895620044&{}:{}&-0.107206358682&{}:{}&1.045310738638&, \\ A^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.097631072938&{}:{}&-0.019672733364&{}:{}&0.922041660427&,\\B^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.087271438453&{}:{}&-0.204988805276&{}:{}&1.117717366823&,\\C^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.290784176704&{}:{}&-0.079459311299&{}:{}&0.788675134595&, \\ A^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.052036359186&{}:{}&0.069158888269&{}:{}&0.878804752545&,\\B^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.030375364847&{}:{}&-0.110808486394&{}:{}&1.080433121547&,\\C^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.124820524074&{}:{}&0.349396052863&{}:{}&0.525783423063&, \\ A^*_{\mathbf{5b}}&{}\approx{}&0.000000000000&{}:{}&0.099900422505&{}:{}&0.900099577495&,\\B^*_{\mathbf{5b}}&{}\approx{}&0.350570754639&{}:{}&0.000000000000&{}:{}&0.649429245361&,\\C^*_{\mathbf{5b}}&{}\approx{}&0.829459311299&{}:{}&0.170540688701&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5b}}}{B^\prime_{\mathbf{5b}}}{C^\prime_{\mathbf{5b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5b}}}{B^{\prime\prime}_{\mathbf{5b}}}{C^{\prime\prime}_{\mathbf{5b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5b}}}{B^{\prime\prime\prime}_{\mathbf{5b}}}{C^{\prime\prime\prime}_{\mathbf{5b}}}\)
\(\triangle{A^*_{\mathbf{5b}}}{B^*_{\mathbf{5b}}}{C^*_{\mathbf{5b}}}\)
5b (303)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5b}}}}&{}\approx{}&1.707106781187&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5b}}}}&{}\approx{}&0.292893218813&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5b}}}}&{}\approx{}&0.078480501450&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.788675134595&{}:{}&-1.366025403784&{}:{}&1.577350269190&,\\ A^\prime_{\mathbf{5b}}&{}\approx{}&0.639245889233&{}:{}&-2.331951230074&{}:{}&2.692705340840&,\\B^\prime_{\mathbf{5b}}&{}\approx{}&0.230997598770&{}:{}&0.307007203691&{}:{}&0.461995197539&,\\C^\prime_{\mathbf{5b}}&{}\approx{}&0.061895620044&{}:{}&-0.107206358682&{}:{}&1.045310738638&. \end{alignedat} \]
5b (303)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5b}}&{}\approx{}&0.073887393036&{}:{}&-0.020190374287&{}:{}&0.946302981250&,\\ A^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.097631072938&{}:{}&-0.019672733364&{}:{}&0.922041660427&,\\B^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.087271438453&{}:{}&-0.204988805276&{}:{}&1.117717366823&,\\C^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.290784176704&{}:{}&-0.079459311299&{}:{}&0.788675134595&. \end{alignedat} \]
5b (303)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5b}}&{}\approx{}&0.025400970204&{}:{}&0.071102078716&{}:{}&0.903496951080&,\\ A^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.052036359186&{}:{}&0.069158888269&{}:{}&0.878804752545&,\\B^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.030375364847&{}:{}&-0.110808486394&{}:{}&1.080433121547&,\\C^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.124820524074&{}:{}&0.349396052863&{}:{}&0.525783423063&. \end{alignedat} \]
5b (303)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5b}}&{}\approx{}&0.116581632223&{}:{}&-0.100577039186&{}:{}&0.983995406963&,\\ A^\prime_{\mathbf{5b}}&{}\approx{}&0.639245889233&{}:{}&-2.331951230074&{}:{}&2.692705340840&,\\B^\prime_{\mathbf{5b}}&{}\approx{}&0.230997598770&{}:{}&0.307007203691&{}:{}&0.461995197539&,\\C^\prime_{\mathbf{5b}}&{}\approx{}&0.061895620044&{}:{}&-0.107206358682&{}:{}&1.045310738638&,\\ A^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.097631072938&{}:{}&-0.019672733364&{}:{}&0.922041660427&,\\B^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.087271438453&{}:{}&-0.204988805276&{}:{}&1.117717366823&,\\C^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.290784176704&{}:{}&-0.079459311299&{}:{}&0.788675134595&, \end{alignedat} \]
5b (303)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5b}}&{}\approx{}&0.327000866942&{}:{}&0.067232897738&{}:{}&0.605766235320&,\\ A^*_{\mathbf{5b}}&{}\approx{}&0.000000000000&{}:{}&0.099900422505&{}:{}&0.900099577495&,\\B^*_{\mathbf{5b}}&{}\approx{}&0.350570754639&{}:{}&0.000000000000&{}:{}&0.649429245361&,\\C^*_{\mathbf{5b}}&{}\approx{}&0.829459311299&{}:{}&0.170540688701&{}:{}&0.000000000000&. \end{alignedat} \]
5b (303)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5b}}&{}\approx{}&0.186111137025&{}:{}&-0.128038238075&{}:{}&0.941927101051&,\\ A^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.097631072938&{}:{}&-0.019672733364&{}:{}&0.922041660427&,\\B^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.087271438453&{}:{}&-0.204988805276&{}:{}&1.117717366823&,\\C^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.290784176704&{}:{}&-0.079459311299&{}:{}&0.788675134595&,\\ A^*_{\mathbf{5b}}&{}\approx{}&0.000000000000&{}:{}&0.099900422505&{}:{}&0.900099577495&,\\B^*_{\mathbf{5b}}&{}\approx{}&0.350570754639&{}:{}&0.000000000000&{}:{}&0.649429245361&,\\C^*_{\mathbf{5b}}&{}\approx{}&0.829459311299&{}:{}&0.170540688701&{}:{}&0.000000000000&. \end{alignedat} \]
5b (303)

Hiroyasu Kamo