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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[Lob & Richmond]

\(\mathbf{2}\) \((020)\)

Malfatti circles

2 (020)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.211324865405&{}:{}&0.366025403784&{}:{}&0.422649730810&, \\ P_{\mathbf{2}}&{}\approx{}&0.002655324025&{}:{}&0.990301631974&{}:{}&0.007043044001&, \\ P^-_{\mathbf{2}}&{}\approx{}&-0.009256832381&{}:{}&1.025939205248&{}:{}&-0.016682372866&, \\ P^+_{\mathbf{2}}&{}\approx{}&0.013346807561&{}:{}&0.958315943161&{}:{}&0.028337249278&, \\ Q_{\mathbf{2}}&{}\approx{}&-0.041319794053&{}:{}&1.171322789857&{}:{}&-0.130002995804&, \\ I^\prime_{\mathbf{2}}&{}\approx{}&0.034820015698&{}:{}&0.884981640663&{}:{}&0.080198343639&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{2}}\)
\(P^-_{\mathbf{2}}\)
\(P^+_{\mathbf{2}}\)
\(Q_{\mathbf{2}}\)
\(I^\prime_{\mathbf{2}}\)
2 (020)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{2}}&{}\approx{}&-0.258819045103&{}:{}&0.584219951998&{}:{}&0.674599093104&,\\B^\prime_{\mathbf{2}}&{}\approx{}&0.011358057904&{}:{}&0.965925826289&{}:{}&0.022716115807&,\\C^\prime_{\mathbf{2}}&{}\approx{}&0.741180954897&{}:{}&1.283763071485&{}:{}&-1.024944026382&, \\ A^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.022373261876&{}:{}&0.970722938164&{}:{}&0.006903799959&,\\B^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.053935377531&{}:{}&0.803005156440&{}:{}&0.143059466029&,\\C^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.002556623083&{}:{}&0.953491170148&{}:{}&0.043952206768&, \\ A^{\prime\prime\prime}_{\mathbf{2}}&{}\approx{}&0.011811902327&{}:{}&1.004522217769&{}:{}&-0.016334120096&,\\B^{\prime\prime\prime}_{\mathbf{2}}&{}\approx{}&1.835145019262&{}:{}&-4.142385791735&{}:{}&3.307240772473&,\\C^{\prime\prime\prime}_{\mathbf{2}}&{}\approx{}&-0.008893874685&{}:{}&0.985712428381&{}:{}&0.023181446304&, \\ A^*_{\mathbf{2}}&{}\approx{}&0.000000000000&{}:{}&1.124844448887&{}:{}&-0.124844448887&,\\B^*_{\mathbf{2}}&{}\approx{}&0.241180954897&{}:{}&0.000000000000&{}:{}&0.758819045103&,\\C^*_{\mathbf{2}}&{}\approx{}&-0.036566092485&{}:{}&1.036566092485&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{2}}}{B^\prime_{\mathbf{2}}}{C^\prime_{\mathbf{2}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{2}}}{B^{\prime\prime}_{\mathbf{2}}}{C^{\prime\prime}_{\mathbf{2}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{2}}}{B^{\prime\prime\prime}_{\mathbf{2}}}{C^{\prime\prime\prime}_{\mathbf{2}}}\)
\(\triangle{A^*_{\mathbf{2}}}{B^*_{\mathbf{2}}}{C^*_{\mathbf{2}}}\)
2 (020)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{2}}}}&{}\approx{}&1.596118591655&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{2}}}}&{}\approx{}&0.053746907075&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{2}}}}&{}\approx{}&3.507305936177&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.211324865405&{}:{}&0.366025403784&{}:{}&0.422649730810&,\\ A^\prime_{\mathbf{2}}&{}\approx{}&-0.258819045103&{}:{}&0.584219951998&{}:{}&0.674599093104&,\\B^\prime_{\mathbf{2}}&{}\approx{}&0.011358057904&{}:{}&0.965925826289&{}:{}&0.022716115807&,\\C^\prime_{\mathbf{2}}&{}\approx{}&0.741180954897&{}:{}&1.283763071485&{}:{}&-1.024944026382&. \end{alignedat} \]
2 (020)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{2}}&{}\approx{}&0.002655324025&{}:{}&0.990301631974&{}:{}&0.007043044001&,\\ A^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.022373261876&{}:{}&0.970722938164&{}:{}&0.006903799959&,\\B^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.053935377531&{}:{}&0.803005156440&{}:{}&0.143059466029&,\\C^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.002556623083&{}:{}&0.953491170148&{}:{}&0.043952206768&. \end{alignedat} \]
2 (020)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{2}}&{}\approx{}&-0.009256832381&{}:{}&1.025939205248&{}:{}&-0.016682372866&,\\ A^{\prime\prime\prime}_{\mathbf{2}}&{}\approx{}&0.011811902327&{}:{}&1.004522217769&{}:{}&-0.016334120096&,\\B^{\prime\prime\prime}_{\mathbf{2}}&{}\approx{}&1.835145019262&{}:{}&-4.142385791735&{}:{}&3.307240772473&,\\C^{\prime\prime\prime}_{\mathbf{2}}&{}\approx{}&-0.008893874685&{}:{}&0.985712428381&{}:{}&0.023181446304&. \end{alignedat} \]
2 (020)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{2}}&{}\approx{}&0.013346807561&{}:{}&0.958315943161&{}:{}&0.028337249278&,\\ A^\prime_{\mathbf{2}}&{}\approx{}&-0.258819045103&{}:{}&0.584219951998&{}:{}&0.674599093104&,\\B^\prime_{\mathbf{2}}&{}\approx{}&0.011358057904&{}:{}&0.965925826289&{}:{}&0.022716115807&,\\C^\prime_{\mathbf{2}}&{}\approx{}&0.741180954897&{}:{}&1.283763071485&{}:{}&-1.024944026382&,\\ A^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.022373261876&{}:{}&0.970722938164&{}:{}&0.006903799959&,\\B^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.053935377531&{}:{}&0.803005156440&{}:{}&0.143059466029&,\\C^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.002556623083&{}:{}&0.953491170148&{}:{}&0.043952206768&, \end{alignedat} \]
2 (020)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{2}}&{}\approx{}&-0.041319794053&{}:{}&1.171322789857&{}:{}&-0.130002995804&,\\ A^*_{\mathbf{2}}&{}\approx{}&0.000000000000&{}:{}&1.124844448887&{}:{}&-0.124844448887&,\\B^*_{\mathbf{2}}&{}\approx{}&0.241180954897&{}:{}&0.000000000000&{}:{}&0.758819045103&,\\C^*_{\mathbf{2}}&{}\approx{}&-0.036566092485&{}:{}&1.036566092485&{}:{}&0.000000000000&. \end{alignedat} \]
2 (020)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{2}}&{}\approx{}&0.034820015698&{}:{}&0.884981640663&{}:{}&0.080198343639&,\\ A^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.022373261876&{}:{}&0.970722938164&{}:{}&0.006903799959&,\\B^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.053935377531&{}:{}&0.803005156440&{}:{}&0.143059466029&,\\C^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.002556623083&{}:{}&0.953491170148&{}:{}&0.043952206768&,\\ A^*_{\mathbf{2}}&{}\approx{}&0.000000000000&{}:{}&1.124844448887&{}:{}&-0.124844448887&,\\B^*_{\mathbf{2}}&{}\approx{}&0.241180954897&{}:{}&0.000000000000&{}:{}&0.758819045103&,\\C^*_{\mathbf{2}}&{}\approx{}&-0.036566092485&{}:{}&1.036566092485&{}:{}&0.000000000000&. \end{alignedat} \]
2 (020)

Hiroyasu Kamo