Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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5a(213)

Malfatti circles

5a (213)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.500000000000&{}:{}&0.666666666667&{}:{}&0.833333333333&, \\ P_{\mathbf{5a}}&{}\approx{}&1.055786112658&{}:{}&-0.001769974007&{}:{}&-0.054016138651&, \\ P^-_{\mathbf{5a}}&{}\approx{}&0.879465324005&{}:{}&0.073985475081&{}:{}&0.046549200915&, \\ P^+_{\mathbf{5a}}&{}\approx{}&1.283786511862&{}:{}&-0.099729339807&{}:{}&-0.184057172055&, \\ Q_{\mathbf{5a}}&{}\approx{}&1.411392400725&{}:{}&0.141552932733&{}:{}&-0.552945333458&, \\ I^\prime_{\mathbf{5a}}&{}\approx{}&1.513520868154&{}:{}&-0.071557185962&{}:{}&-0.441963682191&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{5a}}\)
\(P^-_{\mathbf{5a}}\)
\(P^+_{\mathbf{5a}}\)
\(Q_{\mathbf{5a}}\)
\(I^\prime_{\mathbf{5a}}\)
5a (213)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5a}}&{}\approx{}&1.239614308448&{}:{}&-0.106495248199&{}:{}&-0.133119060249&,\\B^\prime_{\mathbf{5a}}&{}\approx{}&1.565015054523&{}:{}&2.043343369682&{}:{}&-2.608358424204&,\\C^\prime_{\mathbf{5a}}&{}\approx{}&2.048631302823&{}:{}&-2.731508403764&{}:{}&1.682877100941&, \\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.774461866512&{}:{}&-0.024572018155&{}:{}&-0.749889848356&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.269972432037&{}:{}&-0.204998090975&{}:{}&-0.064974341062&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.255414927555&{}:{}&-0.002104641994&{}:{}&-0.253310285560&, \\ A^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&0.319784872146&{}:{}&0.417523330741&{}:{}&0.262691797113&,\\B^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.034105853540&{}:{}&-0.088840019663&{}:{}&0.054734166123&,\\C^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.023815972130&{}:{}&0.086129047986&{}:{}&-0.109945020115&, \\ A^*_{\mathbf{5a}}&{}\approx{}&0.000000000000&{}:{}&-0.344082517041&{}:{}&1.344082517041&,\\B^*_{\mathbf{5a}}&{}\approx{}&1.644122805635&{}:{}&0.000000000000&{}:{}&-0.644122805635&,\\C^*_{\mathbf{5a}}&{}\approx{}&0.908848734284&{}:{}&0.091151265716&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5a}}}{B^\prime_{\mathbf{5a}}}{C^\prime_{\mathbf{5a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5a}}}{B^{\prime\prime}_{\mathbf{5a}}}{C^{\prime\prime}_{\mathbf{5a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5a}}}{B^{\prime\prime\prime}_{\mathbf{5a}}}{C^{\prime\prime\prime}_{\mathbf{5a}}}\)
\(\triangle{A^*_{\mathbf{5a}}}{B^*_{\mathbf{5a}}}{C^*_{\mathbf{5a}}}\)
5a (213)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5a}}}}&{}\approx{}&-0.159742872299&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5a}}}}&{}\approx{}&-3.130030109045&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5a}}}}&{}\approx{}&-4.097262605646&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.500000000000&{}:{}&0.666666666667&{}:{}&0.833333333333&,\\ A^\prime_{\mathbf{5a}}&{}\approx{}&1.239614308448&{}:{}&-0.106495248199&{}:{}&-0.133119060249&,\\B^\prime_{\mathbf{5a}}&{}\approx{}&1.565015054523&{}:{}&2.043343369682&{}:{}&-2.608358424204&,\\C^\prime_{\mathbf{5a}}&{}\approx{}&2.048631302823&{}:{}&-2.731508403764&{}:{}&1.682877100941&. \end{alignedat} \]
5a (213)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5a}}&{}\approx{}&1.055786112658&{}:{}&-0.001769974007&{}:{}&-0.054016138651&,\\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.774461866512&{}:{}&-0.024572018155&{}:{}&-0.749889848356&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.269972432037&{}:{}&-0.204998090975&{}:{}&-0.064974341062&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.255414927555&{}:{}&-0.002104641994&{}:{}&-0.253310285560&. \end{alignedat} \]
5a (213)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5a}}&{}\approx{}&0.879465324005&{}:{}&0.073985475081&{}:{}&0.046549200915&,\\ A^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&0.319784872146&{}:{}&0.417523330741&{}:{}&0.262691797113&,\\B^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.034105853540&{}:{}&-0.088840019663&{}:{}&0.054734166123&,\\C^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.023815972130&{}:{}&0.086129047986&{}:{}&-0.109945020115&. \end{alignedat} \]
5a (213)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5a}}&{}\approx{}&1.283786511862&{}:{}&-0.099729339807&{}:{}&-0.184057172055&,\\ A^\prime_{\mathbf{5a}}&{}\approx{}&1.239614308448&{}:{}&-0.106495248199&{}:{}&-0.133119060249&,\\B^\prime_{\mathbf{5a}}&{}\approx{}&1.565015054523&{}:{}&2.043343369682&{}:{}&-2.608358424204&,\\C^\prime_{\mathbf{5a}}&{}\approx{}&2.048631302823&{}:{}&-2.731508403764&{}:{}&1.682877100941&,\\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.774461866512&{}:{}&-0.024572018155&{}:{}&-0.749889848356&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.269972432037&{}:{}&-0.204998090975&{}:{}&-0.064974341062&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.255414927555&{}:{}&-0.002104641994&{}:{}&-0.253310285560&, \end{alignedat} \]
5a (213)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5a}}&{}\approx{}&1.411392400725&{}:{}&0.141552932733&{}:{}&-0.552945333458&,\\ A^*_{\mathbf{5a}}&{}\approx{}&0.000000000000&{}:{}&-0.344082517041&{}:{}&1.344082517041&,\\B^*_{\mathbf{5a}}&{}\approx{}&1.644122805635&{}:{}&0.000000000000&{}:{}&-0.644122805635&,\\C^*_{\mathbf{5a}}&{}\approx{}&0.908848734284&{}:{}&0.091151265716&{}:{}&0.000000000000&. \end{alignedat} \]
5a (213)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5a}}&{}\approx{}&1.513520868154&{}:{}&-0.071557185962&{}:{}&-0.441963682191&,\\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.774461866512&{}:{}&-0.024572018155&{}:{}&-0.749889848356&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.269972432037&{}:{}&-0.204998090975&{}:{}&-0.064974341062&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.255414927555&{}:{}&-0.002104641994&{}:{}&-0.253310285560&,\\ A^*_{\mathbf{5a}}&{}\approx{}&0.000000000000&{}:{}&-0.344082517041&{}:{}&1.344082517041&,\\B^*_{\mathbf{5a}}&{}\approx{}&1.644122805635&{}:{}&0.000000000000&{}:{}&-0.644122805635&,\\C^*_{\mathbf{5a}}&{}\approx{}&0.908848734284&{}:{}&0.091151265716&{}:{}&0.000000000000&. \end{alignedat} \]
5a (213)