Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


[Top] > The Smallest Pythagorean Triangle > 2c (130)

2c(130)

Malfatti circles

2c (130)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.500000000000&{}:{}&2.000000000000&{}:{}&-2.500000000000&, \\ P_{\mathbf{2c}}&{}\approx{}&0.970637883464&{}:{}&0.288904049605&{}:{}&-0.259541933069&, \\ P^-_{\mathbf{2c}}&{}\approx{}&0.918572418286&{}:{}&0.120609032192&{}:{}&-0.039181450478&, \\ P^+_{\mathbf{2c}}&{}\approx{}&1.014145045823&{}:{}&0.429535437838&{}:{}&-0.443680483662&, \\ Q_{\mathbf{2c}}&{}\approx{}&1.172203012917&{}:{}&-0.522162164214&{}:{}&0.349959151297&, \\ I^\prime_{\mathbf{2c}}&{}\approx{}&1.039077103195&{}:{}&0.654583929966&{}:{}&-0.693661033161&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{2c}}\)
\(P^-_{\mathbf{2c}}\)
\(P^+_{\mathbf{2c}}\)
\(Q_{\mathbf{2c}}\)
\(I^\prime_{\mathbf{2c}}\)
2c (130)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{2c}}&{}\approx{}&1.056565265175&{}:{}&0.226261060701&{}:{}&-0.282826325877&,\\B^\prime_{\mathbf{2c}}&{}\approx{}&6.629510156769&{}:{}&5.419673437846&{}:{}&-11.049183594614&,\\C^\prime_{\mathbf{2c}}&{}\approx{}&0.483616248300&{}:{}&0.644821664400&{}:{}&-0.128437912700&, \\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.901471068418&{}:{}&0.969460335128&{}:{}&-0.870931403546&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.907743518916&{}:{}&0.334980891128&{}:{}&-0.242724410044&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.195655818313&{}:{}&0.355879173613&{}:{}&-0.551534991926&, \\ A^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.644359285343&{}:{}&0.526768467145&{}:{}&-0.171127752489&,\\B^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.853611576376&{}:{}&0.182798987496&{}:{}&-0.036410563872&,\\C^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.157922248310&{}:{}&0.152035788297&{}:{}&-0.309958036606&, \\ A^*_{\mathbf{2c}}&{}\approx{}&0.000000000000&{}:{}&3.032247551123&{}:{}&-2.032247551123&,\\B^*_{\mathbf{2c}}&{}\approx{}&0.770090756738&{}:{}&0.000000000000&{}:{}&0.229909243262&,\\C^*_{\mathbf{2c}}&{}\approx{}&1.803275925284&{}:{}&-0.803275925284&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{2c}}}{B^\prime_{\mathbf{2c}}}{C^\prime_{\mathbf{2c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{2c}}}{B^{\prime\prime}_{\mathbf{2c}}}{C^{\prime\prime}_{\mathbf{2c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{2c}}}{B^{\prime\prime\prime}_{\mathbf{2c}}}{C^{\prime\prime\prime}_{\mathbf{2c}}}\)
\(\triangle{A^*_{\mathbf{2c}}}{B^*_{\mathbf{2c}}}{C^*_{\mathbf{2c}}}\)
2c (130)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{2c}}}}&{}\approx{}&0.113130530351&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{2c}}}}&{}\approx{}&4.419673437846&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{2c}}}}&{}\approx{}&0.322410832200&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.500000000000&{}:{}&2.000000000000&{}:{}&-2.500000000000&,\\ A^\prime_{\mathbf{2c}}&{}\approx{}&1.056565265175&{}:{}&0.226261060701&{}:{}&-0.282826325877&,\\B^\prime_{\mathbf{2c}}&{}\approx{}&6.629510156769&{}:{}&5.419673437846&{}:{}&-11.049183594614&,\\C^\prime_{\mathbf{2c}}&{}\approx{}&0.483616248300&{}:{}&0.644821664400&{}:{}&-0.128437912700&. \end{alignedat} \]
2c (130)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{2c}}&{}\approx{}&0.970637883464&{}:{}&0.288904049605&{}:{}&-0.259541933069&,\\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.901471068418&{}:{}&0.969460335128&{}:{}&-0.870931403546&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.907743518916&{}:{}&0.334980891128&{}:{}&-0.242724410044&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.195655818313&{}:{}&0.355879173613&{}:{}&-0.551534991926&. \end{alignedat} \]
2c (130)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{2c}}&{}\approx{}&0.918572418286&{}:{}&0.120609032192&{}:{}&-0.039181450478&,\\ A^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.644359285343&{}:{}&0.526768467145&{}:{}&-0.171127752489&,\\B^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.853611576376&{}:{}&0.182798987496&{}:{}&-0.036410563872&,\\C^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.157922248310&{}:{}&0.152035788297&{}:{}&-0.309958036606&. \end{alignedat} \]
2c (130)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{2c}}&{}\approx{}&1.014145045823&{}:{}&0.429535437838&{}:{}&-0.443680483662&,\\ A^\prime_{\mathbf{2c}}&{}\approx{}&1.056565265175&{}:{}&0.226261060701&{}:{}&-0.282826325877&,\\B^\prime_{\mathbf{2c}}&{}\approx{}&6.629510156769&{}:{}&5.419673437846&{}:{}&-11.049183594614&,\\C^\prime_{\mathbf{2c}}&{}\approx{}&0.483616248300&{}:{}&0.644821664400&{}:{}&-0.128437912700&,\\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.901471068418&{}:{}&0.969460335128&{}:{}&-0.870931403546&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.907743518916&{}:{}&0.334980891128&{}:{}&-0.242724410044&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.195655818313&{}:{}&0.355879173613&{}:{}&-0.551534991926&, \end{alignedat} \]
2c (130)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{2c}}&{}\approx{}&1.172203012917&{}:{}&-0.522162164214&{}:{}&0.349959151297&,\\ A^*_{\mathbf{2c}}&{}\approx{}&0.000000000000&{}:{}&3.032247551123&{}:{}&-2.032247551123&,\\B^*_{\mathbf{2c}}&{}\approx{}&0.770090756738&{}:{}&0.000000000000&{}:{}&0.229909243262&,\\C^*_{\mathbf{2c}}&{}\approx{}&1.803275925284&{}:{}&-0.803275925284&{}:{}&0.000000000000&. \end{alignedat} \]
2c (130)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{2c}}&{}\approx{}&1.039077103195&{}:{}&0.654583929966&{}:{}&-0.693661033161&,\\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.901471068418&{}:{}&0.969460335128&{}:{}&-0.870931403546&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.907743518916&{}:{}&0.334980891128&{}:{}&-0.242724410044&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.195655818313&{}:{}&0.355879173613&{}:{}&-0.551534991926&,\\ A^*_{\mathbf{2c}}&{}\approx{}&0.000000000000&{}:{}&3.032247551123&{}:{}&-2.032247551123&,\\B^*_{\mathbf{2c}}&{}\approx{}&0.770090756738&{}:{}&0.000000000000&{}:{}&0.229909243262&,\\C^*_{\mathbf{2c}}&{}\approx{}&1.803275925284&{}:{}&-0.803275925284&{}:{}&0.000000000000&. \end{alignedat} \]
2c (130)