Derousseau's Generalization of the Malfatti circles

Pythagorean Triangle U

\(C=90\degree\).   \(a:b:c=5:12:13\).


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2a(031)

Malfatti circles

2a (031)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.250000000000&{}:{}&0.600000000000&{}:{}&0.650000000000&, \\ P_{\mathbf{2a}}&{}\approx{}&-0.008211296765&{}:{}&0.031982673018&{}:{}&0.976228623747&, \\ P^-_{\mathbf{2a}}&{}\approx{}&0.010388126972&{}:{}&-0.011711655184&{}:{}&1.001323528212&, \\ P^+_{\mathbf{2a}}&{}\approx{}&-0.024330763512&{}:{}&0.069851011329&{}:{}&0.954479752183&, \\ Q_{\mathbf{2a}}&{}\approx{}&0.050059173307&{}:{}&-0.270561382752&{}:{}&1.220502209445&, \\ I^\prime_{\mathbf{2a}}&{}\approx{}&-0.050919192507&{}:{}&0.155682082933&{}:{}&0.895237109575&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{2a}}\)
\(P^-_{\mathbf{2a}}\)
\(P^+_{\mathbf{2a}}\)
\(Q_{\mathbf{2a}}\)
\(I^\prime_{\mathbf{2a}}\)
2a (031)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{2a}}&{}\approx{}&0.097501286287&{}:{}&0.433199382582&{}:{}&0.469299331131&,\\B^\prime_{\mathbf{2a}}&{}\approx{}&-1.009078156460&{}:{}&-0.614525050336&{}:{}&2.623603206796&,\\C^\prime_{\mathbf{2a}}&{}\approx{}&-0.018702774091&{}:{}&0.044886657820&{}:{}&0.973816116272&, \\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.036724870524&{}:{}&0.032887186099&{}:{}&1.003837684425&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.007792579159&{}:{}&0.081344656862&{}:{}&0.926447922296&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.070404745655&{}:{}&0.274223673019&{}:{}&0.796181072636&, \\ A^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.019818067163&{}:{}&-0.012069133241&{}:{}&1.031887200405&,\\B^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.009819887146&{}:{}&0.043629876187&{}:{}&0.946550236666&,\\C^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.213408216659&{}:{}&-0.240598084100&{}:{}&1.027189867441&, \\ A^*_{\mathbf{2a}}&{}\approx{}&0.000000000000&{}:{}&-0.284819196258&{}:{}&1.284819196258&,\\B^*_{\mathbf{2a}}&{}\approx{}&0.039399256098&{}:{}&0.000000000000&{}:{}&0.960600743902&,\\C^*_{\mathbf{2a}}&{}\approx{}&-0.227023454471&{}:{}&1.227023454471&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{2a}}}{B^\prime_{\mathbf{2a}}}{C^\prime_{\mathbf{2a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{2a}}}{B^{\prime\prime}_{\mathbf{2a}}}{C^{\prime\prime}_{\mathbf{2a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{2a}}}{B^{\prime\prime\prime}_{\mathbf{2a}}}{C^{\prime\prime\prime}_{\mathbf{2a}}}\)
\(\triangle{A^*_{\mathbf{2a}}}{B^*_{\mathbf{2a}}}{C^*_{\mathbf{2a}}}\)
2a (031)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{2a}}}}&{}\approx{}&0.721998970971&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{2a}}}}&{}\approx{}&4.036312625840&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{2a}}}}&{}\approx{}&0.074811096366&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.250000000000&{}:{}&0.600000000000&{}:{}&0.650000000000&,\\ A^\prime_{\mathbf{2a}}&{}\approx{}&0.097501286287&{}:{}&0.433199382582&{}:{}&0.469299331131&,\\B^\prime_{\mathbf{2a}}&{}\approx{}&-1.009078156460&{}:{}&-0.614525050336&{}:{}&2.623603206796&,\\C^\prime_{\mathbf{2a}}&{}\approx{}&-0.018702774091&{}:{}&0.044886657820&{}:{}&0.973816116272&. \end{alignedat} \]
2a (031)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{2a}}&{}\approx{}&-0.008211296765&{}:{}&0.031982673018&{}:{}&0.976228623747&,\\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.036724870524&{}:{}&0.032887186099&{}:{}&1.003837684425&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.007792579159&{}:{}&0.081344656862&{}:{}&0.926447922296&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.070404745655&{}:{}&0.274223673019&{}:{}&0.796181072636&. \end{alignedat} \]
2a (031)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{2a}}&{}\approx{}&0.010388126972&{}:{}&-0.011711655184&{}:{}&1.001323528212&,\\ A^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.019818067163&{}:{}&-0.012069133241&{}:{}&1.031887200405&,\\B^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.009819887146&{}:{}&0.043629876187&{}:{}&0.946550236666&,\\C^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.213408216659&{}:{}&-0.240598084100&{}:{}&1.027189867441&. \end{alignedat} \]
2a (031)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{2a}}&{}\approx{}&-0.024330763512&{}:{}&0.069851011329&{}:{}&0.954479752183&,\\ A^\prime_{\mathbf{2a}}&{}\approx{}&0.097501286287&{}:{}&0.433199382582&{}:{}&0.469299331131&,\\B^\prime_{\mathbf{2a}}&{}\approx{}&-1.009078156460&{}:{}&-0.614525050336&{}:{}&2.623603206796&,\\C^\prime_{\mathbf{2a}}&{}\approx{}&-0.018702774091&{}:{}&0.044886657820&{}:{}&0.973816116272&,\\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.036724870524&{}:{}&0.032887186099&{}:{}&1.003837684425&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.007792579159&{}:{}&0.081344656862&{}:{}&0.926447922296&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.070404745655&{}:{}&0.274223673019&{}:{}&0.796181072636&, \end{alignedat} \]
2a (031)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{2a}}&{}\approx{}&0.050059173307&{}:{}&-0.270561382752&{}:{}&1.220502209445&,\\ A^*_{\mathbf{2a}}&{}\approx{}&0.000000000000&{}:{}&-0.284819196258&{}:{}&1.284819196258&,\\B^*_{\mathbf{2a}}&{}\approx{}&0.039399256098&{}:{}&0.000000000000&{}:{}&0.960600743902&,\\C^*_{\mathbf{2a}}&{}\approx{}&-0.227023454471&{}:{}&1.227023454471&{}:{}&0.000000000000&. \end{alignedat} \]
2a (031)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{2a}}&{}\approx{}&-0.050919192507&{}:{}&0.155682082933&{}:{}&0.895237109575&,\\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.036724870524&{}:{}&0.032887186099&{}:{}&1.003837684425&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.007792579159&{}:{}&0.081344656862&{}:{}&0.926447922296&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.070404745655&{}:{}&0.274223673019&{}:{}&0.796181072636&,\\ A^*_{\mathbf{2a}}&{}\approx{}&0.000000000000&{}:{}&-0.284819196258&{}:{}&1.284819196258&,\\B^*_{\mathbf{2a}}&{}\approx{}&0.039399256098&{}:{}&0.000000000000&{}:{}&0.960600743902&,\\C^*_{\mathbf{2a}}&{}\approx{}&-0.227023454471&{}:{}&1.227023454471&{}:{}&0.000000000000&. \end{alignedat} \]
2a (031)