# If is isosceles with AB = AC and C (O, r) is the incircle of the touching BC at L, prove that L bisect BC.

Given: If is isosceles with AB = AC and C (O, r) is the incircle of the touching BC at L.

To prove: L bisect BC.

Proof:

Construct the figure according to given condition.

AB = AC (given)

From the theorem which states that the lengths of two tangents drawn from external point to a circle are equal.  ..... (1)

As tangents AP and AQ are drawn from the external point A.

AP = AQ

Also,

AB = AC

⇒ AP + PB = AQ + QC

⇒ AP + PB = AP + QC

⇒ PB =  QC

From (1) as tangents BP and BL are drawn from external point B,

And tangents CQ and CL are drawn from external point C.

⇒ BP = BL  ..... (3)

CQ = CL ...... (4)

As we have proved PB = QC

From 3 and 4

BL = CL

⇒ L bisects BC.

Hence proved.

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