Question -
Answer -
Here chords AB and CDintersect each other at O.
Consider ╬ФAOB and╬ФCOD,
тИаAOB = тИаCOD (They are vertically opposite angles)
OB = OD (Given in thequestion)
OA = OC (Given in thequestion)
So, by SAS congruency,╬ФAOB тЙЕ ╬ФCOD
Also, AB = CD (ByCPCT)
Similarly, ╬ФAOD тЙЕ ╬ФCOB
Or, AD = CB (By CPCT)
In quadrilateral ACBD,opposite sides are equal.
So, ACBD is aparallelogram.
We know that oppositeangles of a parallelogram are equal.
So, тИаA = тИаC
Also, as ABCD is acyclic quadrilateral,
тИаA+тИаC = 180┬░
тЗТтИаA+тИаA = 180┬░
Or, тИаA = 90┬░
As ACBD is aparallelogram and one of its interior angles is 90┬░, so, it is a rectangle.
тИаA is the angle subtended bychord BD. And as тИаA = 90┬░, therefore, BDshould be the diameter of the circle. Similarly, AC is the diameter of thecircle.