Explain the particular rule in △ pqr with the help of given expression : `P/Sinp = Q/Sinq = R/Sinr`

Add the following expressions: p^2-q +r, q^2 -r+ p and r^2-p+q

The triangle PQR of which the angles P,Q,R satisfy cos P = (sin Q)/(2 sin R ) is :

If a,b,c denote the sides of triangleABC , show that the value of the expression, a^3 (p-q)(p-r)+b^2(q-r)+b^2(q-r)(q-p)+c^2(r-p)(r-q) cannot be negative where p,q,r in R .

If PQR is an equilateral triangle inscribed in the auxiliary circle of the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1(a>b) and P'Q'R' is corresponding triangle inscribed with the ellipse then centroid of the triangle P'Q'R' lies at