Let O,P,Q,R,S (O being origin) be five points all lying on a circle in order. P,Q,R,S have respective position vector p,q,r,s

Question

samigupta8 · Answer

∆POQ.∆ROS + ∆POS.QOR =?
A).∆PRQ ∆PRS
B).∆POR ∆QOS
C).([]OPQR)([]QQRS)
D).([]PQRS)^2
{[] denotes AREA of quadrilateral AND ∆denotes area of triangle}

samigupta8 · Answer

I m just not getting this .Can we prove this part?
(p*q)•(r*s)+(p*s)•(q*r)=(p*r)•(q*s)

parthkohli · Answer

Assume that $P_i = \left(r\cos \theta_i, r \sin \theta_i \right)$ and then check.

samigupta8 · Answer

Really! That way.

parthkohli · Answer

Yeah. That's a vector by the way.

samigupta8 · Answer

Yep,that's a vector and do you intend to say that i should solve for each one of them individually (like taking dot product of each of them).

parthkohli · Answer

Unfortunately...

samigupta8 · Answer

I think that it is really very long way to go with.
Didn't try that(obviously).
I thought some STP would work out here in better way.

parthkohli · Answer

I haven't studied vector geometry or stuff.
Here's another way:
PQRS is a cyclic quadrilateral. What do we know about cyclic quadrilaterals? :D

samigupta8 · Answer

Not much ..(the geometry part is really very boring to me)
Just this,the opposite angles are supplementary.

parthkohli · Answer

Yeah. I don't know... check if that works, maybe?

samigupta8 · Answer

@ganeshie8