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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[Lob & Richmond]

\(\mathbf{4}\) \((200)\)

Malfatti circles

4 (200)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.214285714286&{}:{}&0.321428571429&{}:{}&0.464285714286&, \\ P_{\mathbf{4}}&{}\approx{}&0.999371961038&{}:{}&0.000184450286&{}:{}&0.000443588676&, \\ P^-_{\mathbf{4}}&{}\approx{}&1.013062489312&{}:{}&-0.005417484204&{}:{}&-0.007645005107&, \\ P^+_{\mathbf{4}}&{}\approx{}&0.986142818874&{}:{}&0.005597593320&{}:{}&0.008259587806&, \\ Q_{\mathbf{4}}&{}\approx{}&1.083158670498&{}:{}&-0.018613504890&{}:{}&-0.064545165608&, \\ I^\prime_{\mathbf{4}}&{}\approx{}&0.954516994553&{}:{}&0.015882006906&{}:{}&0.029600998541&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{4}}\)
\(P^-_{\mathbf{4}}\)
\(P^+_{\mathbf{4}}\)
\(Q_{\mathbf{4}}\)
\(I^\prime_{\mathbf{4}}\)
4 (200)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4}}&{}\approx{}&0.986655667145&{}:{}&0.005459045259&{}:{}&0.007885287596&,\\B^\prime_{\mathbf{4}}&{}\approx{}&0.582449053990&{}:{}&-0.844422004303&{}:{}&1.261972950312&,\\C^\prime_{\mathbf{4}}&{}\approx{}&2.202096923453&{}:{}&3.303145385179&{}:{}&-4.505242308632&, \\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.921234039768&{}:{}&0.023132965856&{}:{}&0.055632994376&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.988661089360&{}:{}&0.010900076171&{}:{}&0.000438834469&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.984145713436&{}:{}&0.000181640035&{}:{}&0.015672646529&, \\ A^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&-0.400694693995&{}:{}&0.580918475699&{}:{}&0.819776218296&,\\B^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&1.002017614697&{}:{}&0.005544041038&{}:{}&-0.007561655735&,\\C^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&0.997362670103&{}:{}&-0.005333527367&{}:{}&0.007970857264&, \\ A^*_{\mathbf{4}}&{}\approx{}&0.000000000000&{}:{}&0.223831198584&{}:{}&0.776168801416&,\\B^*_{\mathbf{4}}&{}\approx{}&1.063365707697&{}:{}&0.000000000000&{}:{}&-0.063365707697&,\\C^*_{\mathbf{4}}&{}\approx{}&1.017484936752&{}:{}&-0.017484936752&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4}}}{B^\prime_{\mathbf{4}}}{C^\prime_{\mathbf{4}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4}}}{B^{\prime\prime}_{\mathbf{4}}}{C^{\prime\prime}_{\mathbf{4}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4}}}{B^{\prime\prime\prime}_{\mathbf{4}}}{C^{\prime\prime\prime}_{\mathbf{4}}}\)
\(\triangle{A^*_{\mathbf{4}}}{B^*_{\mathbf{4}}}{C^*_{\mathbf{4}}}\)
4 (200)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4}}}}&{}\approx{}&0.016983696362&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4}}}}&{}\approx{}&2.718095585288&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4}}}}&{}\approx{}&10.276452309447&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.214285714286&{}:{}&0.321428571429&{}:{}&0.464285714286&,\\ A^\prime_{\mathbf{4}}&{}\approx{}&0.986655667145&{}:{}&0.005459045259&{}:{}&0.007885287596&,\\B^\prime_{\mathbf{4}}&{}\approx{}&0.582449053990&{}:{}&-0.844422004303&{}:{}&1.261972950312&,\\C^\prime_{\mathbf{4}}&{}\approx{}&2.202096923453&{}:{}&3.303145385179&{}:{}&-4.505242308632&. \end{alignedat} \]
4 (200)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4}}&{}\approx{}&0.999371961038&{}:{}&0.000184450286&{}:{}&0.000443588676&,\\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.921234039768&{}:{}&0.023132965856&{}:{}&0.055632994376&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.988661089360&{}:{}&0.010900076171&{}:{}&0.000438834469&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.984145713436&{}:{}&0.000181640035&{}:{}&0.015672646529&. \end{alignedat} \]
4 (200)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4}}&{}\approx{}&1.013062489312&{}:{}&-0.005417484204&{}:{}&-0.007645005107&,\\ A^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&-0.400694693995&{}:{}&0.580918475699&{}:{}&0.819776218296&,\\B^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&1.002017614697&{}:{}&0.005544041038&{}:{}&-0.007561655735&,\\C^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&0.997362670103&{}:{}&-0.005333527367&{}:{}&0.007970857264&. \end{alignedat} \]
4 (200)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4}}&{}\approx{}&0.986142818874&{}:{}&0.005597593320&{}:{}&0.008259587806&,\\ A^\prime_{\mathbf{4}}&{}\approx{}&0.986655667145&{}:{}&0.005459045259&{}:{}&0.007885287596&,\\B^\prime_{\mathbf{4}}&{}\approx{}&0.582449053990&{}:{}&-0.844422004303&{}:{}&1.261972950312&,\\C^\prime_{\mathbf{4}}&{}\approx{}&2.202096923453&{}:{}&3.303145385179&{}:{}&-4.505242308632&,\\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.921234039768&{}:{}&0.023132965856&{}:{}&0.055632994376&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.988661089360&{}:{}&0.010900076171&{}:{}&0.000438834469&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.984145713436&{}:{}&0.000181640035&{}:{}&0.015672646529&, \end{alignedat} \]
4 (200)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4}}&{}\approx{}&1.083158670498&{}:{}&-0.018613504890&{}:{}&-0.064545165608&,\\ A^*_{\mathbf{4}}&{}\approx{}&0.000000000000&{}:{}&0.223831198584&{}:{}&0.776168801416&,\\B^*_{\mathbf{4}}&{}\approx{}&1.063365707697&{}:{}&0.000000000000&{}:{}&-0.063365707697&,\\C^*_{\mathbf{4}}&{}\approx{}&1.017484936752&{}:{}&-0.017484936752&{}:{}&0.000000000000&. \end{alignedat} \]
4 (200)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4}}&{}\approx{}&0.954516994553&{}:{}&0.015882006906&{}:{}&0.029600998541&,\\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.921234039768&{}:{}&0.023132965856&{}:{}&0.055632994376&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.988661089360&{}:{}&0.010900076171&{}:{}&0.000438834469&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.984145713436&{}:{}&0.000181640035&{}:{}&0.015672646529&,\\ A^*_{\mathbf{4}}&{}\approx{}&0.000000000000&{}:{}&0.223831198584&{}:{}&0.776168801416&,\\B^*_{\mathbf{4}}&{}\approx{}&1.063365707697&{}:{}&0.000000000000&{}:{}&-0.063365707697&,\\C^*_{\mathbf{4}}&{}\approx{}&1.017484936752&{}:{}&-0.017484936752&{}:{}&0.000000000000&. \end{alignedat} \]
4 (200)

Hiroyasu Kamo