Question

If bisectors of opposite angles of a cyclic quadrilateral ABCD intersect the circle, circumscribing it at the point P and Q, prove that PQ is a diameter of the circle.

Answer 1

Hint: Use the property of cyclic quadrilateral is supplementary that is sum of opposite sides of a cyclic quadrilateral is $180{}^\circ $. Further, simplify it to prove the required result.

Complete step by step answer:
We are given that ABCD is a cyclic quadrilateral. A cyclic quadrilateral is a quadrilateral which has all its four vertices lying on a circle that means a quadrilateral which is circumscribed in a circle is known as cyclic quadrilateral. DP and QB are the bisectors of angle D and B respectively. We have to prove that PQ is the diameter of a circle. For proving this we will do a construction of joining the points QD and QC.

Since ABCD is a cyclic quadrilateral so by the angle sum property of opposite angles of cyclic quadrilateral we have the sum of opposite angles of cyclic quadrilateral is $180{}^\circ $ . That is, $\angle CDA+\angle CBA=180{}^\circ $
On dividing both sides by 2, we get
$\dfrac{1}{2}\angle CDA+\dfrac{1}{2}\angle CBA=\dfrac{180}{2}=90{}^\circ $
In the figure it can be seen that $\angle 1=\dfrac{1}{2}\angle CDA$ and $\angle 2=\dfrac{1}{2}\angle CBA$ as DP and QB are the bisectors of angle D and B respectively. So the above equation now becomes
$\angle 1+\angle 2=90{}^\circ .......(1)$
We know that the angles in the same segment QC are equal. Thus $\angle 2=\angle 3$ . Then the equation (1) becomes
$\angle 1+\angle 3=90{}^\circ $
That means angle $\angle 1+\angle 3=\angle PDQ=90{}^\circ $. Thus PDQ is a right angled triangle inside the circle which implies the hypotenuse will be the diameter of the circle. Hence PQ is the diameter of the circle because diameter of the circle subtends a right angle at the circumference.

Note: Since ABCD is a cyclic quadrilateral so the sum of opposite angles of cyclic quadrilateral is $180{}^\circ $ . That is$\angle CDA+\angle CBA=180{}^\circ $. PQ is the diameter of the circle because diameter of the circle subtends a right angle at the circumference.