Question 531229
Solution: 
Let OA = a
 OB = b,
 OC = c,
OD = d,
 OE = e,
 OF = f,
 [OAB] =x,
 [OCD] = y, 
[OEF] = z,
 [ODE] = u,
[OFA] = v 
[OBC]= w. 
We are given that v^2 = zx, w^2 = xy and we have to prove that u^2 =yz.
Since∠AOB =∠DOE, we have
u\x =(1/2 de sin∠DOE)/(1/2 ab sin∠AOB)= de/ab

Similarly, we obtain
v/y =fa/cd
w/z =bc/ef
Multiplying, these three equalities, we get uvw =xyz. Hence
x^2 * y^2 * z^2 =u^2*v^2*w^2 =u^2 (zx)(xy).
This gives u^2=yz, as desired.