Derousseau's Generalization of the Malfatti circles

Test Case in ETC

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

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


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\(\mathbf{0a}\) \((011)\)

Malfatti circles

0a (011)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.375000000000&{}:{}&0.562500000000&{}:{}&0.812500000000&, \\ P_{\mathbf{a}}&{}\approx{}&-0.207564625291&{}:{}&0.699126789355&{}:{}&0.508437835935&, \\ P^-_{\mathbf{a}}&{}\approx{}&-0.159542460854&{}:{}&0.738312738897&{}:{}&0.421229721957&, \\ P^+_{\mathbf{a}}&{}\approx{}&-0.238081621429&{}:{}&0.674225007337&{}:{}&0.563856614092&, \\ Q_{\mathbf{a}}&{}\approx{}&-0.117842720415&{}:{}&0.809222742208&{}:{}&0.308619978206&, \\ I^\prime_{\mathbf{a}}&{}\approx{}&-0.281570361744&{}:{}&0.632898214354&{}:{}&0.648672147390&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{a}}\)
\(P^-_{\mathbf{a}}\)
\(P^+_{\mathbf{a}}\)
\(Q_{\mathbf{a}}\)
\(I^\prime_{\mathbf{a}}\)
0a (011)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{a}}&{}\approx{}&-0.679420184127&{}:{}&0.687035529870&{}:{}&0.992384654257&,\\B^\prime_{\mathbf{a}}&{}\approx{}&-0.124513520549&{}:{}&0.854734226027&{}:{}&0.269779294522&,\\C^\prime_{\mathbf{a}}&{}\approx{}&-0.226425905618&{}:{}&0.339638858427&{}:{}&0.886787047191&, \\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.160672096378&{}:{}&0.671978078225&{}:{}&0.488694018153&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.376283018429&{}:{}&0.454562743015&{}:{}&0.921720275413&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.243121193007&{}:{}&0.818889725805&{}:{}&0.424231467202&, \\ A^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.102238559843&{}:{}&0.701825761030&{}:{}&0.400412798813&,\\B^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.377152916947&{}:{}&0.381380271282&{}:{}&0.995772645665&,\\C^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.196528416585&{}:{}&0.909472204096&{}:{}&0.287056212489&, \\ A^*_{\mathbf{a}}&{}\approx{}&0.000000000000&{}:{}&0.723914668343&{}:{}&0.276085331657&,\\B^*_{\mathbf{a}}&{}\approx{}&-0.617697946698&{}:{}&0.000000000000&{}:{}&1.617697946698&,\\C^*_{\mathbf{a}}&{}\approx{}&-0.170445654633&{}:{}&1.170445654633&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{a}}}{B^\prime_{\mathbf{a}}}{C^\prime_{\mathbf{a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{a}}}{B^{\prime\prime}_{\mathbf{a}}}{C^{\prime\prime}_{\mathbf{a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{a}}}{B^{\prime\prime\prime}_{\mathbf{a}}}{C^{\prime\prime\prime}_{\mathbf{a}}}\)
\(\triangle{A^*_{\mathbf{a}}}{B^*_{\mathbf{a}}}{C^*_{\mathbf{a}}}\)
0a (011)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{a}}}}&{}\approx{}&1.221396497547&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{a}}}}&{}\approx{}&0.332036054796&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{a}}}}&{}\approx{}&0.603802414982&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.375000000000&{}:{}&0.562500000000&{}:{}&0.812500000000&,\\ A^\prime_{\mathbf{a}}&{}\approx{}&-0.679420184127&{}:{}&0.687035529870&{}:{}&0.992384654257&,\\B^\prime_{\mathbf{a}}&{}\approx{}&-0.124513520549&{}:{}&0.854734226027&{}:{}&0.269779294522&,\\C^\prime_{\mathbf{a}}&{}\approx{}&-0.226425905618&{}:{}&0.339638858427&{}:{}&0.886787047191&. \end{alignedat} \]
0a (011)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{a}}&{}\approx{}&-0.207564625291&{}:{}&0.699126789355&{}:{}&0.508437835935&,\\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.160672096378&{}:{}&0.671978078225&{}:{}&0.488694018153&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.376283018429&{}:{}&0.454562743015&{}:{}&0.921720275413&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.243121193007&{}:{}&0.818889725805&{}:{}&0.424231467202&. \end{alignedat} \]
0a (011)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{a}}&{}\approx{}&-0.159542460854&{}:{}&0.738312738897&{}:{}&0.421229721957&,\\ A^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.102238559843&{}:{}&0.701825761030&{}:{}&0.400412798813&,\\B^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.377152916947&{}:{}&0.381380271282&{}:{}&0.995772645665&,\\C^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.196528416585&{}:{}&0.909472204096&{}:{}&0.287056212489&. \end{alignedat} \]
0a (011)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{a}}&{}\approx{}&-0.238081621429&{}:{}&0.674225007337&{}:{}&0.563856614092&,\\ A^\prime_{\mathbf{a}}&{}\approx{}&-0.679420184127&{}:{}&0.687035529870&{}:{}&0.992384654257&,\\B^\prime_{\mathbf{a}}&{}\approx{}&-0.124513520549&{}:{}&0.854734226027&{}:{}&0.269779294522&,\\C^\prime_{\mathbf{a}}&{}\approx{}&-0.226425905618&{}:{}&0.339638858427&{}:{}&0.886787047191&,\\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.160672096378&{}:{}&0.671978078225&{}:{}&0.488694018153&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.376283018429&{}:{}&0.454562743015&{}:{}&0.921720275413&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.243121193007&{}:{}&0.818889725805&{}:{}&0.424231467202&, \end{alignedat} \]
0a (011)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{a}}&{}\approx{}&-0.117842720415&{}:{}&0.809222742208&{}:{}&0.308619978206&,\\ A^*_{\mathbf{a}}&{}\approx{}&0.000000000000&{}:{}&0.723914668343&{}:{}&0.276085331657&,\\B^*_{\mathbf{a}}&{}\approx{}&-0.617697946698&{}:{}&0.000000000000&{}:{}&1.617697946698&,\\C^*_{\mathbf{a}}&{}\approx{}&-0.170445654633&{}:{}&1.170445654633&{}:{}&0.000000000000&. \end{alignedat} \]
0a (011)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{a}}&{}\approx{}&-0.281570361744&{}:{}&0.632898214354&{}:{}&0.648672147390&,\\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.160672096378&{}:{}&0.671978078225&{}:{}&0.488694018153&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.376283018429&{}:{}&0.454562743015&{}:{}&0.921720275413&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.243121193007&{}:{}&0.818889725805&{}:{}&0.424231467202&,\\ A^*_{\mathbf{a}}&{}\approx{}&0.000000000000&{}:{}&0.723914668343&{}:{}&0.276085331657&,\\B^*_{\mathbf{a}}&{}\approx{}&-0.617697946698&{}:{}&0.000000000000&{}:{}&1.617697946698&,\\C^*_{\mathbf{a}}&{}\approx{}&-0.170445654633&{}:{}&1.170445654633&{}:{}&0.000000000000&. \end{alignedat} \]
0a (011)

Hiroyasu Kamo