MathDB

Let

ABCD

be a cyclic quadrilateral with circumcentre

O,

and the pair of opposite sides not parallel with each other. Let

M=AB\cap CD

and

N=AD\cap BC.

Denote, by

P,Q,S,T;

the intersection of the internal angle bisectors of

\angle MAN

and

\angle MBN;

\angle MBN

and

\angle MCN;

\angle MDN

and

\angle MAN;

\angle MCN

and

\angle MDN.

Suppose that the four points

P,Q,S,T

are distinct. (a) Show that the four points

P,Q,S,T

are concyclic. Find the centre of this circle, and denote it as

I.

(b) Let

E=AC\cap BD.

Prove that

E,O,I

are collinear.

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