Derousseau's Generalization of the Malfatti circles

Test Case in ETC

The test case used in “Encyclopedis of Triangle Centers” to search for triangle centers.

\(a:b:c=6:9:13\).


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2b(121)

Malfatti circles

2b (121)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.600000000000&{}:{}&-0.900000000000&{}:{}&1.300000000000&, \\ P_{\mathbf{2b}}&{}\approx{}&-0.003716837056&{}:{}&1.264740884515&{}:{}&-0.261024047459&, \\ P^-_{\mathbf{2b}}&{}\approx{}&0.147712151652&{}:{}&0.721763609679&{}:{}&0.130524238668&, \\ P^+_{\mathbf{2b}}&{}\approx{}&-0.307581015132&{}:{}&2.354303356929&{}:{}&-1.046722341797&, \\ Q_{\mathbf{2b}}&{}\approx{}&-1.033919930423&{}:{}&5.097261414898&{}:{}&-3.063341484475&, \\ I^\prime_{\mathbf{2b}}&{}\approx{}&-0.108026788254&{}:{}&2.440569118297&{}:{}&-1.332542330043&, \end{alignedat} \]
\(I_{\mathbf{b}}\) Incenter
\(P_{\mathbf{2b}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{2b}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{2b}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{2b}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{2b}}\) Radical center of the Malfatti circles
2b (121)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{2b}}&{}\approx{}&3.117538670626&{}:{}&4.764462008909&{}:{}&-6.882000679535&,\\B^\prime_{\mathbf{2b}}&{}\approx{}&-0.613794451455&{}:{}&2.943682429607&{}:{}&-1.329887978152&,\\C^\prime_{\mathbf{2b}}&{}\approx{}&-0.731373904386&{}:{}&1.097060856579&{}:{}&0.634313047807&, \\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.667445442798&{}:{}&2.101077063119&{}:{}&-0.433631620321&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.010999451647&{}:{}&1.783463066709&{}:{}&-0.772463615061&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.009517497835&{}:{}&3.238551609440&{}:{}&-2.229034111606&, \\ A^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.214446567756&{}:{}&1.028459270193&{}:{}&0.185987297563&,\\B^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.293090456776&{}:{}&0.447923344027&{}:{}&0.258986199197&,\\C^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.271829716307&{}:{}&1.328237352618&{}:{}&-0.600067068924&, \\ A^*_{\mathbf{2b}}&{}\approx{}&0.000000000000&{}:{}&2.506126882702&{}:{}&-1.506126882702&,\\B^*_{\mathbf{2b}}&{}\approx{}&0.252344145449&{}:{}&0.000000000000&{}:{}&0.747655854551&,\\C^*_{\mathbf{2b}}&{}\approx{}&-0.254450661943&{}:{}&1.254450661943&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{2b}}}{B^\prime_{\mathbf{2b}}}{C^\prime_{\mathbf{2b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{2b}}}{B^{\prime\prime}_{\mathbf{2b}}}{C^{\prime\prime}_{\mathbf{2b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{2b}}}{B^{\prime\prime\prime}_{\mathbf{2b}}}{C^{\prime\prime\prime}_{\mathbf{2b}}}\)
\(\triangle{A^*_{\mathbf{2b}}}{B^*_{\mathbf{2b}}}{C^*_{\mathbf{2b}}}\)
2b (121)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{2b}}}}&{}\approx{}&-5.293846676565&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{2b}}}}&{}\approx{}&-1.022990752424&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{2b}}}}&{}\approx{}&-1.218956507310&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.600000000000&{}:{}&-0.900000000000&{}:{}&1.300000000000&,\\ A^\prime_{\mathbf{2b}}&{}\approx{}&3.117538670626&{}:{}&4.764462008909&{}:{}&-6.882000679535&,\\B^\prime_{\mathbf{2b}}&{}\approx{}&-0.613794451455&{}:{}&2.943682429607&{}:{}&-1.329887978152&,\\C^\prime_{\mathbf{2b}}&{}\approx{}&-0.731373904386&{}:{}&1.097060856579&{}:{}&0.634313047807&. \end{alignedat} \]
2b (121)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{2b}}&{}\approx{}&-0.003716837056&{}:{}&1.264740884515&{}:{}&-0.261024047459&,\\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.667445442798&{}:{}&2.101077063119&{}:{}&-0.433631620321&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.010999451647&{}:{}&1.783463066709&{}:{}&-0.772463615061&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.009517497835&{}:{}&3.238551609440&{}:{}&-2.229034111606&. \end{alignedat} \]
2b (121)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{2b}}&{}\approx{}&0.147712151652&{}:{}&0.721763609679&{}:{}&0.130524238668&,\\ A^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.214446567756&{}:{}&1.028459270193&{}:{}&0.185987297563&,\\B^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.293090456776&{}:{}&0.447923344027&{}:{}&0.258986199197&,\\C^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.271829716307&{}:{}&1.328237352618&{}:{}&-0.600067068924&. \end{alignedat} \]
2b (121)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{2b}}&{}\approx{}&-0.307581015132&{}:{}&2.354303356929&{}:{}&-1.046722341797&,\\ A^\prime_{\mathbf{2b}}&{}\approx{}&3.117538670626&{}:{}&4.764462008909&{}:{}&-6.882000679535&,\\B^\prime_{\mathbf{2b}}&{}\approx{}&-0.613794451455&{}:{}&2.943682429607&{}:{}&-1.329887978152&,\\C^\prime_{\mathbf{2b}}&{}\approx{}&-0.731373904386&{}:{}&1.097060856579&{}:{}&0.634313047807&,\\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.667445442798&{}:{}&2.101077063119&{}:{}&-0.433631620321&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.010999451647&{}:{}&1.783463066709&{}:{}&-0.772463615061&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.009517497835&{}:{}&3.238551609440&{}:{}&-2.229034111606&, \end{alignedat} \]
2b (121)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{2b}}&{}\approx{}&-1.033919930423&{}:{}&5.097261414898&{}:{}&-3.063341484475&,\\ A^*_{\mathbf{2b}}&{}\approx{}&0.000000000000&{}:{}&2.506126882702&{}:{}&-1.506126882702&,\\B^*_{\mathbf{2b}}&{}\approx{}&0.252344145449&{}:{}&0.000000000000&{}:{}&0.747655854551&,\\C^*_{\mathbf{2b}}&{}\approx{}&-0.254450661943&{}:{}&1.254450661943&{}:{}&0.000000000000&. \end{alignedat} \]
2b (121)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{2b}}&{}\approx{}&-0.108026788254&{}:{}&2.440569118297&{}:{}&-1.332542330043&,\\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.667445442798&{}:{}&2.101077063119&{}:{}&-0.433631620321&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.010999451647&{}:{}&1.783463066709&{}:{}&-0.772463615061&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.009517497835&{}:{}&3.238551609440&{}:{}&-2.229034111606&,\\ A^*_{\mathbf{2b}}&{}\approx{}&0.000000000000&{}:{}&2.506126882702&{}:{}&-1.506126882702&,\\B^*_{\mathbf{2b}}&{}\approx{}&0.252344145449&{}:{}&0.000000000000&{}:{}&0.747655854551&,\\C^*_{\mathbf{2b}}&{}\approx{}&-0.254450661943&{}:{}&1.254450661943&{}:{}&0.000000000000&. \end{alignedat} \]
2b (121)