Derousseau's Generalization of the Malfatti circles

Isosceles Triangle with 120° Top Angle

\(A=30\degree\), \(B=30\degree\), \(C=120\degree\).


[Top] > Isosceles Triangle with 120° Top Angle > 7a (233)

7a(233)

Malfatti circles

7a (233)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.577350269190&{}:{}&0.577350269190&{}:{}&1.000000000000&, \\ P_{\mathbf{7a}}&{}\approx{}&1.001311947531&{}:{}&-0.000733657657&{}:{}&-0.000578289874&, \\ P^-_{\mathbf{7a}}&{}\approx{}&0.981700717908&{}:{}&0.006447699134&{}:{}&0.011851582958&, \\ P^+_{\mathbf{7a}}&{}\approx{}&1.021422839873&{}:{}&-0.008097983917&{}:{}&-0.013324855957&, \\ Q_{\mathbf{7a}}&{}\approx{}&0.880051842567&{}:{}&0.088903289864&{}:{}&0.031044867569&, \\ I^\prime_{\mathbf{7a}}&{}\approx{}&1.062356189989&{}:{}&-0.028756245388&{}:{}&-0.033599944601&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{7a}}\)
\(P^-_{\mathbf{7a}}\)
\(P^+_{\mathbf{7a}}\)
\(Q_{\mathbf{7a}}\)
\(I^\prime_{\mathbf{7a}}\)
7a (233)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7a}}&{}\approx{}&1.020366712256&{}:{}&-0.007454734077&{}:{}&-0.012911978179&,\\B^\prime_{\mathbf{7a}}&{}\approx{}&5.990582397203&{}:{}&5.385410681680&{}:{}&-10.375993078883&,\\C^\prime_{\mathbf{7a}}&{}\approx{}&0.605171715523&{}:{}&-0.605171715523&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.099292352121&{}:{}&-0.055525539462&{}:{}&-0.043766812659&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.015314419308&{}:{}&-0.014728042556&{}:{}&-0.000586376752&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.026544005831&{}:{}&-0.000752145095&{}:{}&-0.025791860736&, \\ A^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.281578854170&{}:{}&0.253133613468&{}:{}&0.465287532361&,\\B^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.995256047475&{}:{}&-0.007271277163&{}:{}&0.012015229688&,\\C^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.006123636603&{}:{}&0.006608106098&{}:{}&-0.012731742701&, \\ A^*_{\mathbf{7a}}&{}\approx{}&0.000000000000&{}:{}&0.741180954897&{}:{}&0.258819045103&,\\B^*_{\mathbf{7a}}&{}\approx{}&0.965925826289&{}:{}&0.000000000000&{}:{}&0.034074173711&,\\C^*_{\mathbf{7a}}&{}\approx{}&0.908248290464&{}:{}&0.091751709536&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7a}}}{B^\prime_{\mathbf{7a}}}{C^\prime_{\mathbf{7a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7a}}}{B^{\prime\prime}_{\mathbf{7a}}}{C^{\prime\prime}_{\mathbf{7a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7a}}}{B^{\prime\prime\prime}_{\mathbf{7a}}}{C^{\prime\prime\prime}_{\mathbf{7a}}}\)
\(\triangle{A^*_{\mathbf{7a}}}{B^*_{\mathbf{7a}}}{C^*_{\mathbf{7a}}}\)
7a (233)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7a}}}}&{}\approx{}&-0.012911978179&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7a}}}}&{}\approx{}&-10.375993078883&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7a}}}}&{}\approx{}&-1.048188158589&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.577350269190&{}:{}&0.577350269190&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{7a}}&{}\approx{}&1.020366712256&{}:{}&-0.007454734077&{}:{}&-0.012911978179&,\\B^\prime_{\mathbf{7a}}&{}\approx{}&5.990582397203&{}:{}&5.385410681680&{}:{}&-10.375993078883&,\\C^\prime_{\mathbf{7a}}&{}\approx{}&0.605171715523&{}:{}&-0.605171715523&{}:{}&1.000000000000&. \end{alignedat} \]
7a (233)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7a}}&{}\approx{}&1.001311947531&{}:{}&-0.000733657657&{}:{}&-0.000578289874&,\\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.099292352121&{}:{}&-0.055525539462&{}:{}&-0.043766812659&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.015314419308&{}:{}&-0.014728042556&{}:{}&-0.000586376752&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.026544005831&{}:{}&-0.000752145095&{}:{}&-0.025791860736&. \end{alignedat} \]
7a (233)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7a}}&{}\approx{}&0.981700717908&{}:{}&0.006447699134&{}:{}&0.011851582958&,\\ A^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.281578854170&{}:{}&0.253133613468&{}:{}&0.465287532361&,\\B^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.995256047475&{}:{}&-0.007271277163&{}:{}&0.012015229688&,\\C^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.006123636603&{}:{}&0.006608106098&{}:{}&-0.012731742701&. \end{alignedat} \]
7a (233)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7a}}&{}\approx{}&1.021422839873&{}:{}&-0.008097983917&{}:{}&-0.013324855957&,\\ A^\prime_{\mathbf{7a}}&{}\approx{}&1.020366712256&{}:{}&-0.007454734077&{}:{}&-0.012911978179&,\\B^\prime_{\mathbf{7a}}&{}\approx{}&5.990582397203&{}:{}&5.385410681680&{}:{}&-10.375993078883&,\\C^\prime_{\mathbf{7a}}&{}\approx{}&0.605171715523&{}:{}&-0.605171715523&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.099292352121&{}:{}&-0.055525539462&{}:{}&-0.043766812659&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.015314419308&{}:{}&-0.014728042556&{}:{}&-0.000586376752&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.026544005831&{}:{}&-0.000752145095&{}:{}&-0.025791860736&, \end{alignedat} \]
7a (233)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7a}}&{}\approx{}&0.880051842567&{}:{}&0.088903289864&{}:{}&0.031044867569&,\\ A^*_{\mathbf{7a}}&{}\approx{}&0.000000000000&{}:{}&0.741180954897&{}:{}&0.258819045103&,\\B^*_{\mathbf{7a}}&{}\approx{}&0.965925826289&{}:{}&0.000000000000&{}:{}&0.034074173711&,\\C^*_{\mathbf{7a}}&{}\approx{}&0.908248290464&{}:{}&0.091751709536&{}:{}&0.000000000000&. \end{alignedat} \]
7a (233)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7a}}&{}\approx{}&1.062356189989&{}:{}&-0.028756245388&{}:{}&-0.033599944601&,\\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.099292352121&{}:{}&-0.055525539462&{}:{}&-0.043766812659&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.015314419308&{}:{}&-0.014728042556&{}:{}&-0.000586376752&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.026544005831&{}:{}&-0.000752145095&{}:{}&-0.025791860736&,\\ A^*_{\mathbf{7a}}&{}\approx{}&0.000000000000&{}:{}&0.741180954897&{}:{}&0.258819045103&,\\B^*_{\mathbf{7a}}&{}\approx{}&0.965925826289&{}:{}&0.000000000000&{}:{}&0.034074173711&,\\C^*_{\mathbf{7a}}&{}\approx{}&0.908248290464&{}:{}&0.091751709536&{}:{}&0.000000000000&. \end{alignedat} \]
7a (233)