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 > 7b (323)

7b (323)

Malfatti circles

7b (323)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.577350269190&{}:{}&-0.577350269190&{}:{}&1.000000000000&, \\ P_{\mathbf{7b}}&{}\approx{}&-0.000733657657&{}:{}&1.001311947531&{}:{}&-0.000578289874&, \\ P^-_{\mathbf{7b}}&{}\approx{}&0.006447699134&{}:{}&0.981700717908&{}:{}&0.011851582958&, \\ P^+_{\mathbf{7b}}&{}\approx{}&-0.008097983917&{}:{}&1.021422839873&{}:{}&-0.013324855957&, \\ Q_{\mathbf{7b}}&{}\approx{}&0.088903289864&{}:{}&0.880051842567&{}:{}&0.031044867569&, \\ I^\prime_{\mathbf{7b}}&{}\approx{}&-0.028756245388&{}:{}&1.062356189989&{}:{}&-0.033599944601&, \end{alignedat} \]
\(I_{\mathbf{b}}\) Incenter
\(P_{\mathbf{7b}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{7b}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{7b}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{7b}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{7b}}\) Radical center of the Malfatti circles
7b (323)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7b}}&{}\approx{}&5.385410681680&{}:{}&5.990582397203&{}:{}&-10.375993078883&,\\B^\prime_{\mathbf{7b}}&{}\approx{}&-0.007454734077&{}:{}&1.020366712256&{}:{}&-0.012911978179&,\\C^\prime_{\mathbf{7b}}&{}\approx{}&-0.605171715523&{}:{}&0.605171715523&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.014728042556&{}:{}&1.015314419308&{}:{}&-0.000586376752&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.055525539462&{}:{}&1.099292352121&{}:{}&-0.043766812659&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.000752145095&{}:{}&1.026544005831&{}:{}&-0.025791860736&, \\ A^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.007271277163&{}:{}&0.995256047475&{}:{}&0.012015229688&,\\B^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.253133613468&{}:{}&0.281578854170&{}:{}&0.465287532361&,\\C^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.006608106098&{}:{}&1.006123636603&{}:{}&-0.012731742701&, \\ A^*_{\mathbf{7b}}&{}\approx{}&0.000000000000&{}:{}&0.965925826289&{}:{}&0.034074173711&,\\B^*_{\mathbf{7b}}&{}\approx{}&0.741180954897&{}:{}&0.000000000000&{}:{}&0.258819045103&,\\C^*_{\mathbf{7b}}&{}\approx{}&0.091751709536&{}:{}&0.908248290464&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7b}}}{B^\prime_{\mathbf{7b}}}{C^\prime_{\mathbf{7b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7b}}}{B^{\prime\prime}_{\mathbf{7b}}}{C^{\prime\prime}_{\mathbf{7b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7b}}}{B^{\prime\prime\prime}_{\mathbf{7b}}}{C^{\prime\prime\prime}_{\mathbf{7b}}}\)
\(\triangle{A^*_{\mathbf{7b}}}{B^*_{\mathbf{7b}}}{C^*_{\mathbf{7b}}}\)
7b (323)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7b}}}}&{}\approx{}&-10.375993078883&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7b}}}}&{}\approx{}&-0.012911978179&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7b}}}}&{}\approx{}&-1.048188158589&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.577350269190&{}:{}&-0.577350269190&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{7b}}&{}\approx{}&5.385410681680&{}:{}&5.990582397203&{}:{}&-10.375993078883&,\\B^\prime_{\mathbf{7b}}&{}\approx{}&-0.007454734077&{}:{}&1.020366712256&{}:{}&-0.012911978179&,\\C^\prime_{\mathbf{7b}}&{}\approx{}&-0.605171715523&{}:{}&0.605171715523&{}:{}&1.000000000000&. \end{alignedat} \]
7b (323)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7b}}&{}\approx{}&-0.000733657657&{}:{}&1.001311947531&{}:{}&-0.000578289874&,\\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.014728042556&{}:{}&1.015314419308&{}:{}&-0.000586376752&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.055525539462&{}:{}&1.099292352121&{}:{}&-0.043766812659&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.000752145095&{}:{}&1.026544005831&{}:{}&-0.025791860736&. \end{alignedat} \]
7b (323)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7b}}&{}\approx{}&0.006447699134&{}:{}&0.981700717908&{}:{}&0.011851582958&,\\ A^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.007271277163&{}:{}&0.995256047475&{}:{}&0.012015229688&,\\B^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.253133613468&{}:{}&0.281578854170&{}:{}&0.465287532361&,\\C^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.006608106098&{}:{}&1.006123636603&{}:{}&-0.012731742701&. \end{alignedat} \]
7b (323)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7b}}&{}\approx{}&-0.008097983917&{}:{}&1.021422839873&{}:{}&-0.013324855957&,\\ A^\prime_{\mathbf{7b}}&{}\approx{}&5.385410681680&{}:{}&5.990582397203&{}:{}&-10.375993078883&,\\B^\prime_{\mathbf{7b}}&{}\approx{}&-0.007454734077&{}:{}&1.020366712256&{}:{}&-0.012911978179&,\\C^\prime_{\mathbf{7b}}&{}\approx{}&-0.605171715523&{}:{}&0.605171715523&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.014728042556&{}:{}&1.015314419308&{}:{}&-0.000586376752&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.055525539462&{}:{}&1.099292352121&{}:{}&-0.043766812659&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.000752145095&{}:{}&1.026544005831&{}:{}&-0.025791860736&, \end{alignedat} \]
7b (323)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7b}}&{}\approx{}&0.088903289864&{}:{}&0.880051842567&{}:{}&0.031044867569&,\\ A^*_{\mathbf{7b}}&{}\approx{}&0.000000000000&{}:{}&0.965925826289&{}:{}&0.034074173711&,\\B^*_{\mathbf{7b}}&{}\approx{}&0.741180954897&{}:{}&0.000000000000&{}:{}&0.258819045103&,\\C^*_{\mathbf{7b}}&{}\approx{}&0.091751709536&{}:{}&0.908248290464&{}:{}&0.000000000000&. \end{alignedat} \]
7b (323)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7b}}&{}\approx{}&-0.028756245388&{}:{}&1.062356189989&{}:{}&-0.033599944601&,\\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.014728042556&{}:{}&1.015314419308&{}:{}&-0.000586376752&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.055525539462&{}:{}&1.099292352121&{}:{}&-0.043766812659&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.000752145095&{}:{}&1.026544005831&{}:{}&-0.025791860736&,\\ A^*_{\mathbf{7b}}&{}\approx{}&0.000000000000&{}:{}&0.965925826289&{}:{}&0.034074173711&,\\B^*_{\mathbf{7b}}&{}\approx{}&0.741180954897&{}:{}&0.000000000000&{}:{}&0.258819045103&,\\C^*_{\mathbf{7b}}&{}\approx{}&0.091751709536&{}:{}&0.908248290464&{}:{}&0.000000000000&. \end{alignedat} \]
7b (323)