MathDB

3

Part of 1997 APMO

Problems(1)

I am finding another solution! new ideas!!

Source: 1997 APMO

9/14/2004
Let ABCABC be a triangle inscribed in a circle and let la=maMa ,  lb=mbMb ,  lc=mcMc , l_a = \frac{m_a}{M_a} \ , \ \ l_b = \frac{m_b}{M_b} \ , \ \ l_c = \frac{m_c}{M_c} \ , where mam_a,mbm_b, mcm_c are the lengths of the angle bisectors (internal to the triangle) and MaM_a, MbM_b, McM_c are the lengths of the angle bisectors extended until they meet the circle. Prove that lasin⁡2A+lbsin⁡2B+lcsin⁡2C≥3 \frac{l_a}{\sin^2 A} + \frac{l_b}{\sin^2 B} + \frac{l_c}{\sin^2 C} \geq 3 and that equality holds iff ABCABC is an equilateral triangle.
trigonometrygeometrycircumcircleinequalitiesratioangle bisectorgeometric inequality