if in a triangle PQR,$\displaystyle \sin P,\sin Q,\sin R$ are in A.P,then
which of the following is true
(A)the altitudes are in A.P
(B)the altitudes are in H.P
(C)the medians are in G.P
(D)the medians are in A.P
let p,q and r be the sides of the triangle PQR.
then according the properties of triangle
p/sinP = q/sinQ = r/sinR = k. So
p = k*sinP, q = k*sinQ and r = k*sinR. and 2q = p + r
Let L1, L2 and L3 be the altitudes from P. Q and R points,
then the area of the triangle PQR is given by
1/2*p*L1 = 1/2*q*L2 = 1/2*r*L3 = A
So p = 2A/L1, q = 2A/L2 and r = 2A/L3.
Now can you proceed?