ABC is a given triangle. Forces P,Q,R acting along the lines OA,CB,OC are in equilibrium. Show that P:Q:R=a^2(b^2+c^2-a^2):b^2(c^2+a^2-b^2):c^2(a^2+b^2-c^2) if O is the circumcentre of the triangle

By Lami‟s theorem, P/sin(BOC) = Q/sin(COA) = R/sin(AOB)OR, P/sin(2A) = Q/sin(2B) = R/sin(2C)Or, P/2sinA.cosA = Q/2sinB.cosB = R/2sinC.cosC ………………………….(1)WE HAVE, cos A = (b^2 + c^2 – a^2)/2bcsinA = 2.(area)/bcso, 2sinAcosA = 2(2.(area)/bc).( (b^2 + c^2 – a^2)/2bc) =2.(area)( b^2 + c^2 – a^2)/b^2 .c^2Similarly we find 2sinB.cosB and 2sinC.cosC and putting these values in (1) and dividing it by a^2.b^2.c^2, we get the required result