Question

In the given circle, O is the centre and AD, AE are the two tangents. If BC is also a tangent, then prove that 2AE = AB + BC + AC

Answer 1

From the figure,
AE = AB + BE

BE = BP (Tangents from an external point to a circle are equal in length)

Therefore, AE = AB + BP …(i)

Also, AE = AD (Tangents from an external point to a circle are equal in length)

And, AD = AC + CD

But CD = CP (Tangents from an external point to a circle are equal in length)

Therefore, AE = AC + CP ….(ii)

On adding the equations (i) and (ii), we get,

2AE = AB + BP + AC + CP

Since, BP+CP = BC,
2AE = AB + BC + AC