Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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4a(211)

Malfatti circles

4a (211)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.500000000000&{}:{}&0.666666666667&{}:{}&0.833333333333&, \\ P_{\mathbf{4a}}&{}\approx{}&1.066840293917&{}:{}&-0.012258601144&{}:{}&-0.054581692773&, \\ P^-_{\mathbf{4a}}&{}\approx{}&0.670912547922&{}:{}&0.159300269295&{}:{}&0.169787182783&, \\ P^+_{\mathbf{4a}}&{}\approx{}&1.867314760466&{}:{}&-0.359111017238&{}:{}&-0.508203743228&, \\ Q_{\mathbf{4a}}&{}\approx{}&2.893047208112&{}:{}&-0.759629632407&{}:{}&-1.133417575705&, \\ I^\prime_{\mathbf{4a}}&{}\approx{}&1.711705482154&{}:{}&-0.211869851448&{}:{}&-0.499835630705&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{4a}}\)
\(P^-_{\mathbf{4a}}\)
\(P^+_{\mathbf{4a}}\)
\(Q_{\mathbf{4a}}\)
\(I^\prime_{\mathbf{4a}}\)
4a (211)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4a}}&{}\approx{}&2.298631302823&{}:{}&-0.577169467921&{}:{}&-0.721461834902&,\\B^\prime_{\mathbf{4a}}&{}\approx{}&1.683049043273&{}:{}&2.122032695515&{}:{}&-2.805081738788&,\\C^\prime_{\mathbf{4a}}&{}\approx{}&0.989614308448&{}:{}&-1.319485744597&{}:{}&1.329871436149&, \\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.246374268551&{}:{}&-0.045185377164&{}:{}&-0.201188891387&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.900285500615&{}:{}&-0.803063076633&{}:{}&-0.097222423981&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.172695213497&{}:{}&-0.024965502505&{}:{}&-1.147729710992&, \\ A^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&0.277419102863&{}:{}&0.349777333577&{}:{}&0.372803563560&,\\B^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&0.998027544212&{}:{}&-0.250597399399&{}:{}&0.252569855186&,\\C^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.082752187239&{}:{}&0.257086732900&{}:{}&-0.339838920139&, \\ A^*_{\mathbf{4a}}&{}\approx{}&0.000000000000&{}:{}&0.401273475459&{}:{}&0.598726524541&,\\B^*_{\mathbf{4a}}&{}\approx{}&1.644122805635&{}:{}&0.000000000000&{}:{}&-0.644122805635&,\\C^*_{\mathbf{4a}}&{}\approx{}&1.356062329784&{}:{}&-0.356062329784&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4a}}}{B^\prime_{\mathbf{4a}}}{C^\prime_{\mathbf{4a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4a}}}{B^{\prime\prime}_{\mathbf{4a}}}{C^{\prime\prime}_{\mathbf{4a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4a}}}{B^{\prime\prime\prime}_{\mathbf{4a}}}{C^{\prime\prime\prime}_{\mathbf{4a}}}\)
\(\triangle{A^*_{\mathbf{4a}}}{B^*_{\mathbf{4a}}}{C^*_{\mathbf{4a}}}\)
4a (211)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4a}}}}&{}\approx{}&-0.865754201882&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4a}}}}&{}\approx{}&-3.366098086545&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4a}}}}&{}\approx{}&-1.979228616896&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.500000000000&{}:{}&0.666666666667&{}:{}&0.833333333333&,\\ A^\prime_{\mathbf{4a}}&{}\approx{}&2.298631302823&{}:{}&-0.577169467921&{}:{}&-0.721461834902&,\\B^\prime_{\mathbf{4a}}&{}\approx{}&1.683049043273&{}:{}&2.122032695515&{}:{}&-2.805081738788&,\\C^\prime_{\mathbf{4a}}&{}\approx{}&0.989614308448&{}:{}&-1.319485744597&{}:{}&1.329871436149&. \end{alignedat} \]
4a (211)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4a}}&{}\approx{}&1.066840293917&{}:{}&-0.012258601144&{}:{}&-0.054581692773&,\\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.246374268551&{}:{}&-0.045185377164&{}:{}&-0.201188891387&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.900285500615&{}:{}&-0.803063076633&{}:{}&-0.097222423981&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.172695213497&{}:{}&-0.024965502505&{}:{}&-1.147729710992&. \end{alignedat} \]
4a (211)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4a}}&{}\approx{}&0.670912547922&{}:{}&0.159300269295&{}:{}&0.169787182783&,\\ A^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&0.277419102863&{}:{}&0.349777333577&{}:{}&0.372803563560&,\\B^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&0.998027544212&{}:{}&-0.250597399399&{}:{}&0.252569855186&,\\C^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.082752187239&{}:{}&0.257086732900&{}:{}&-0.339838920139&. \end{alignedat} \]
4a (211)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4a}}&{}\approx{}&1.867314760466&{}:{}&-0.359111017238&{}:{}&-0.508203743228&,\\ A^\prime_{\mathbf{4a}}&{}\approx{}&2.298631302823&{}:{}&-0.577169467921&{}:{}&-0.721461834902&,\\B^\prime_{\mathbf{4a}}&{}\approx{}&1.683049043273&{}:{}&2.122032695515&{}:{}&-2.805081738788&,\\C^\prime_{\mathbf{4a}}&{}\approx{}&0.989614308448&{}:{}&-1.319485744597&{}:{}&1.329871436149&,\\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.246374268551&{}:{}&-0.045185377164&{}:{}&-0.201188891387&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.900285500615&{}:{}&-0.803063076633&{}:{}&-0.097222423981&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.172695213497&{}:{}&-0.024965502505&{}:{}&-1.147729710992&, \end{alignedat} \]
4a (211)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4a}}&{}\approx{}&2.893047208112&{}:{}&-0.759629632407&{}:{}&-1.133417575705&,\\ A^*_{\mathbf{4a}}&{}\approx{}&0.000000000000&{}:{}&0.401273475459&{}:{}&0.598726524541&,\\B^*_{\mathbf{4a}}&{}\approx{}&1.644122805635&{}:{}&0.000000000000&{}:{}&-0.644122805635&,\\C^*_{\mathbf{4a}}&{}\approx{}&1.356062329784&{}:{}&-0.356062329784&{}:{}&0.000000000000&. \end{alignedat} \]
4a (211)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4a}}&{}\approx{}&1.711705482154&{}:{}&-0.211869851448&{}:{}&-0.499835630705&,\\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.246374268551&{}:{}&-0.045185377164&{}:{}&-0.201188891387&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.900285500615&{}:{}&-0.803063076633&{}:{}&-0.097222423981&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.172695213497&{}:{}&-0.024965502505&{}:{}&-1.147729710992&,\\ A^*_{\mathbf{4a}}&{}\approx{}&0.000000000000&{}:{}&0.401273475459&{}:{}&0.598726524541&,\\B^*_{\mathbf{4a}}&{}\approx{}&1.644122805635&{}:{}&0.000000000000&{}:{}&-0.644122805635&,\\C^*_{\mathbf{4a}}&{}\approx{}&1.356062329784&{}:{}&-0.356062329784&{}:{}&0.000000000000&. \end{alignedat} \]
4a (211)