In the given figure, if AB = AC then prove that BQ = CQ.

As, from point A, AP and AR are the tangents to the circle.

We know that,

If two tangents are drawn to a circle from an exterior point, the tangents are equal in length.

So, we have AP = AR ……….(1)

From point B, BP and BQ are the tangents to the circle.

∴ BP = BQ ……….(2)

From point C, CQ and CR are the tangents to the circle.

∴ CQ = CR …………(3)

Adding equations (1), (2) and (3), we get :

⇒ AP + BP + CQ = AR + BQ + CR

⇒ (AP + BP) + CQ = (AR + CR) + BQ

⇒ AB + CQ = AC + BQ

Given,

AB = AC

∴ BQ = CQ.

Hence, proved that BQ = CQ.