If `a=q+r+s,b=r+s-p,c=p+q+r` then prove that `a^2+b^2+c^2-2ab-2ac+2bc=r^2`

if p: q :: r , prove that p : r = p^(2): q^(2) .

If P + Q = R and P - Q = S, prove that R^2 + S^2 = 2(P^2 + Q^2)

If P + Q = R and P - Q = S, prove that R^2 + S^2 = 2(P^2 + Q^2)

Ifa,b,c,p,q,r be six complex numbers such that a/p+b/q+c/r=0 and p/a+q/b+r/c=1-i then prove that (p^2/a^2)+(q^2/b^2)+(r^2/c^2)=-2i

If p + r = 2q and 1/q + 1/s = 2/r , then prove that p : q = r : s .

AB and AC are two chords of a circle of radius r such that AB = 2AC. If p and q are the distances of AB and AC from the centre, prove that 4q^2 = p^2 + 3r^2

If a sectheta+bt a ntheta+c=0 and psectheta+qtantheta+r=0, prove that (b r-q c)^2-(p c-a r)^2=(a q-b p)^2

If p, q, r, s are in G.P, show that (p^2 + q^2 + r^2) (q^2 + r^2 + s^2) = (pq + qr + rs)^2

If p,q and r are in AP,then prove that (p+2q-r)(2q+r-p)(r+p-q)=4pqr

Given A=[[p,b,ca,q,ca,b,r]]. If det(A)=0 and a!=p,b!=q,c!=q, then prove p-a,+(q)/(q-b)+(r)/(r-c)=2