Derousseau's Generalization of the Malfatti circles

Isosceles Triangle with 90° Top Angle

\(A=45\degree\), \(B=45\degree\), \(C=90\degree\).


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7(222)

Malfatti circles

7 (222)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.292893218813&{}:{}&0.292893218813&{}:{}&0.414213562373&, \\ P_{\mathbf{7}}&{}\approx{}&0.477208273403&{}:{}&0.477208273403&{}:{}&0.045583453195&, \\ P^-_{\mathbf{7}}&{}\approx{}&0.189425282639&{}:{}&0.189425282639&{}:{}&0.621149434723&, \\ P^+_{\mathbf{7}}&{}\approx{}&0.341636102194&{}:{}&0.341636102194&{}:{}&0.316727795612&, \\ Q_{\mathbf{7}}&{}\approx{}&0.403190537839&{}:{}&0.403190537839&{}:{}&0.193618924323&, \\ I^\prime_{\mathbf{7}}&{}\approx{}&0.422378967344&{}:{}&0.422378967344&{}:{}&0.155242065312&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{7}}\)
\(P^-_{\mathbf{7}}\)
\(P^+_{\mathbf{7}}\)
\(Q_{\mathbf{7}}\)
\(I^\prime_{\mathbf{7}}\)
7 (222)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7}}&{}\approx{}&-0.207106781187&{}:{}&0.500000000000&{}:{}&0.707106781187&,\\B^\prime_{\mathbf{7}}&{}\approx{}&0.500000000000&{}:{}&-0.207106781187&{}:{}&0.707106781187&,\\C^\prime_{\mathbf{7}}&{}\approx{}&1.558260083544&{}:{}&1.558260083544&{}:{}&-2.116520167087&, \\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.757076375334&{}:{}&0.221742536459&{}:{}&0.021181088207&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.221742536459&{}:{}&0.757076375334&{}:{}&0.021181088207&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.146446609407&{}:{}&0.146446609407&{}:{}&0.707106781187&, \\ A^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&-1.294545023646&{}:{}&0.536218105897&{}:{}&1.758326917749&,\\B^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&0.536218105897&{}:{}&-1.294545023646&{}:{}&1.758326917749&,\\C^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&-0.707106781187&{}:{}&-0.707106781187&{}:{}&2.414213562373&, \\ A^*_{\mathbf{7}}&{}\approx{}&0.000000000000&{}:{}&0.675576651179&{}:{}&0.324423348821&,\\B^*_{\mathbf{7}}&{}\approx{}&0.675576651179&{}:{}&0.000000000000&{}:{}&0.324423348821&,\\C^*_{\mathbf{7}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7}}}{B^\prime_{\mathbf{7}}}{C^\prime_{\mathbf{7}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7}}}{B^{\prime\prime}_{\mathbf{7}}}{C^{\prime\prime}_{\mathbf{7}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7}}}{B^{\prime\prime\prime}_{\mathbf{7}}}{C^{\prime\prime\prime}_{\mathbf{7}}}\)
\(\triangle{A^*_{\mathbf{7}}}{B^*_{\mathbf{7}}}{C^*_{\mathbf{7}}}\)
7 (222)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7}}}}&{}\approx{}&1.707106781187&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7}}}}&{}\approx{}&1.707106781187&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7}}}}&{}\approx{}&5.320232710939&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.292893218813&{}:{}&0.292893218813&{}:{}&0.414213562373&,\\ A^\prime_{\mathbf{7}}&{}\approx{}&-0.207106781187&{}:{}&0.500000000000&{}:{}&0.707106781187&,\\B^\prime_{\mathbf{7}}&{}\approx{}&0.500000000000&{}:{}&-0.207106781187&{}:{}&0.707106781187&,\\C^\prime_{\mathbf{7}}&{}\approx{}&1.558260083544&{}:{}&1.558260083544&{}:{}&-2.116520167087&. \end{alignedat} \]
7 (222)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7}}&{}\approx{}&0.477208273403&{}:{}&0.477208273403&{}:{}&0.045583453195&,\\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.757076375334&{}:{}&0.221742536459&{}:{}&0.021181088207&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.221742536459&{}:{}&0.757076375334&{}:{}&0.021181088207&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.146446609407&{}:{}&0.146446609407&{}:{}&0.707106781187&. \end{alignedat} \]
7 (222)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7}}&{}\approx{}&0.189425282639&{}:{}&0.189425282639&{}:{}&0.621149434723&,\\ A^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&-1.294545023646&{}:{}&0.536218105897&{}:{}&1.758326917749&,\\B^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&0.536218105897&{}:{}&-1.294545023646&{}:{}&1.758326917749&,\\C^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&-0.707106781187&{}:{}&-0.707106781187&{}:{}&2.414213562373&. \end{alignedat} \]
7 (222)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7}}&{}\approx{}&0.341636102194&{}:{}&0.341636102194&{}:{}&0.316727795612&,\\ A^\prime_{\mathbf{7}}&{}\approx{}&-0.207106781187&{}:{}&0.500000000000&{}:{}&0.707106781187&,\\B^\prime_{\mathbf{7}}&{}\approx{}&0.500000000000&{}:{}&-0.207106781187&{}:{}&0.707106781187&,\\C^\prime_{\mathbf{7}}&{}\approx{}&1.558260083544&{}:{}&1.558260083544&{}:{}&-2.116520167087&,\\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.757076375334&{}:{}&0.221742536459&{}:{}&0.021181088207&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.221742536459&{}:{}&0.757076375334&{}:{}&0.021181088207&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.146446609407&{}:{}&0.146446609407&{}:{}&0.707106781187&, \end{alignedat} \]
7 (222)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7}}&{}\approx{}&0.403190537839&{}:{}&0.403190537839&{}:{}&0.193618924323&,\\ A^*_{\mathbf{7}}&{}\approx{}&0.000000000000&{}:{}&0.675576651179&{}:{}&0.324423348821&,\\B^*_{\mathbf{7}}&{}\approx{}&0.675576651179&{}:{}&0.000000000000&{}:{}&0.324423348821&,\\C^*_{\mathbf{7}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
7 (222)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7}}&{}\approx{}&0.422378967344&{}:{}&0.422378967344&{}:{}&0.155242065312&,\\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.757076375334&{}:{}&0.221742536459&{}:{}&0.021181088207&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.221742536459&{}:{}&0.757076375334&{}:{}&0.021181088207&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.146446609407&{}:{}&0.146446609407&{}:{}&0.707106781187&,\\ A^*_{\mathbf{7}}&{}\approx{}&0.000000000000&{}:{}&0.675576651179&{}:{}&0.324423348821&,\\B^*_{\mathbf{7}}&{}\approx{}&0.675576651179&{}:{}&0.000000000000&{}:{}&0.324423348821&,\\C^*_{\mathbf{7}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
7 (222)