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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\(\mathbf{6c}\) \((330)\)

Malfatti circles

6c (330)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.000000000000&{}:{}&4.500000000000&{}:{}&-6.500000000000&, \\ P_{\mathbf{6c}}&{}\approx{}&0.409119242429&{}:{}&0.835793160362&{}:{}&-0.244912402791&, \\ P^-_{\mathbf{6c}}&{}\approx{}&0.264904257589&{}:{}&0.631834127402&{}:{}&0.103261615009&, \\ P^+_{\mathbf{6c}}&{}\approx{}&0.538887740715&{}:{}&1.019320950130&{}:{}&-0.558208690846&, \\ Q_{\mathbf{6c}}&{}\approx{}&0.385750953027&{}:{}&0.435884543852&{}:{}&0.178364503120&, \\ I^\prime_{\mathbf{6c}}&{}\approx{}&0.620479807063&{}:{}&1.086169829001&{}:{}&-0.706649636064&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{6c}}\)
\(P^-_{\mathbf{6c}}\)
\(P^+_{\mathbf{6c}}\)
\(Q_{\mathbf{6c}}\)
\(I^\prime_{\mathbf{6c}}\)
6c (330)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6c}}&{}\approx{}&1.724870757207&{}:{}&1.630959203716&{}:{}&-2.355829960922&,\\B^\prime_{\mathbf{6c}}&{}\approx{}&0.382109304120&{}:{}&1.445794188140&{}:{}&-0.827903492259&,\\C^\prime_{\mathbf{6c}}&{}\approx{}&0.320679435390&{}:{}&0.481019153086&{}:{}&0.198301411524&, \\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.348709616457&{}:{}&0.921241792019&{}:{}&-0.269951408477&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.640494701591&{}:{}&0.742926756197&{}:{}&-0.383421457788&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.731283288886&{}:{}&1.493944815475&{}:{}&-1.225228104361&, \\ A^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.185113271195&{}:{}&0.700416579120&{}:{}&0.114470149684&,\\B^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.428199186483&{}:{}&0.404885642185&{}:{}&0.166915171332&,\\C^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.495211159428&{}:{}&1.181148667237&{}:{}&-0.676359826665&, \\ A^*_{\mathbf{6c}}&{}\approx{}&0.000000000000&{}:{}&0.709621848012&{}:{}&0.290378151988&,\\B^*_{\mathbf{6c}}&{}\approx{}&0.683815606936&{}:{}&0.000000000000&{}:{}&0.316184393064&,\\C^*_{\mathbf{6c}}&{}\approx{}&0.469491586589&{}:{}&0.530508413411&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6c}}}{B^\prime_{\mathbf{6c}}}{C^\prime_{\mathbf{6c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6c}}}{B^{\prime\prime}_{\mathbf{6c}}}{C^{\prime\prime}_{\mathbf{6c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6c}}}{B^{\prime\prime\prime}_{\mathbf{6c}}}{C^{\prime\prime\prime}_{\mathbf{6c}}}\)
\(\triangle{A^*_{\mathbf{6c}}}{B^*_{\mathbf{6c}}}{C^*_{\mathbf{6c}}}\)
6c (330)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6c}}}}&{}\approx{}&0.362435378603&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6c}}}}&{}\approx{}&0.127369768040&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6c}}}}&{}\approx{}&0.106893145130&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.000000000000&{}:{}&4.500000000000&{}:{}&-6.500000000000&,\\ A^\prime_{\mathbf{6c}}&{}\approx{}&1.724870757207&{}:{}&1.630959203716&{}:{}&-2.355829960922&,\\B^\prime_{\mathbf{6c}}&{}\approx{}&0.382109304120&{}:{}&1.445794188140&{}:{}&-0.827903492259&,\\C^\prime_{\mathbf{6c}}&{}\approx{}&0.320679435390&{}:{}&0.481019153086&{}:{}&0.198301411524&. \end{alignedat} \]
6c (330)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6c}}&{}\approx{}&0.409119242429&{}:{}&0.835793160362&{}:{}&-0.244912402791&,\\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.348709616457&{}:{}&0.921241792019&{}:{}&-0.269951408477&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.640494701591&{}:{}&0.742926756197&{}:{}&-0.383421457788&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.731283288886&{}:{}&1.493944815475&{}:{}&-1.225228104361&. \end{alignedat} \]
6c (330)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6c}}&{}\approx{}&0.264904257589&{}:{}&0.631834127402&{}:{}&0.103261615009&,\\ A^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.185113271195&{}:{}&0.700416579120&{}:{}&0.114470149684&,\\B^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.428199186483&{}:{}&0.404885642185&{}:{}&0.166915171332&,\\C^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.495211159428&{}:{}&1.181148667237&{}:{}&-0.676359826665&. \end{alignedat} \]
6c (330)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6c}}&{}\approx{}&0.538887740715&{}:{}&1.019320950130&{}:{}&-0.558208690846&,\\ A^\prime_{\mathbf{6c}}&{}\approx{}&1.724870757207&{}:{}&1.630959203716&{}:{}&-2.355829960922&,\\B^\prime_{\mathbf{6c}}&{}\approx{}&0.382109304120&{}:{}&1.445794188140&{}:{}&-0.827903492259&,\\C^\prime_{\mathbf{6c}}&{}\approx{}&0.320679435390&{}:{}&0.481019153086&{}:{}&0.198301411524&,\\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.348709616457&{}:{}&0.921241792019&{}:{}&-0.269951408477&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.640494701591&{}:{}&0.742926756197&{}:{}&-0.383421457788&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.731283288886&{}:{}&1.493944815475&{}:{}&-1.225228104361&, \end{alignedat} \]
6c (330)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6c}}&{}\approx{}&0.385750953027&{}:{}&0.435884543852&{}:{}&0.178364503120&,\\ A^*_{\mathbf{6c}}&{}\approx{}&0.000000000000&{}:{}&0.709621848012&{}:{}&0.290378151988&,\\B^*_{\mathbf{6c}}&{}\approx{}&0.683815606936&{}:{}&0.000000000000&{}:{}&0.316184393064&,\\C^*_{\mathbf{6c}}&{}\approx{}&0.469491586589&{}:{}&0.530508413411&{}:{}&0.000000000000&. \end{alignedat} \]
6c (330)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6c}}&{}\approx{}&0.620479807063&{}:{}&1.086169829001&{}:{}&-0.706649636064&,\\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.348709616457&{}:{}&0.921241792019&{}:{}&-0.269951408477&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.640494701591&{}:{}&0.742926756197&{}:{}&-0.383421457788&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.731283288886&{}:{}&1.493944815475&{}:{}&-1.225228104361&,\\ A^*_{\mathbf{6c}}&{}\approx{}&0.000000000000&{}:{}&0.709621848012&{}:{}&0.290378151988&,\\B^*_{\mathbf{6c}}&{}\approx{}&0.683815606936&{}:{}&0.000000000000&{}:{}&0.316184393064&,\\C^*_{\mathbf{6c}}&{}\approx{}&0.469491586589&{}:{}&0.530508413411&{}:{}&0.000000000000&. \end{alignedat} \]
6c (330)

Hiroyasu Kamo