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

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#### Solution

Let us first put the given data in the form of a diagram.

It is given that triangle *ABC* is isosceles with

*AB = AC …*… (1)

By looking at the figure we can rewrite the above equation as,

*AM + MB = AN + NC*

From the property of tangents we know that the length of two tangents drawn to a circle from the same external point will be equal. Therefore,

*AM = AN*

Let us substitute *AN* with *AM* in the equation (1). We get,

*AM + MB = AM + NC*

*MB = NC* …… (2)

From the property of tangents we know that the length of two tangents drawn from the same external point will be equal. Therefore we have,

*MB = BL*

*NC = LC*

But from equation (2), we have found that

*MB = NC*

Therefore,

*BL = LC*

Thus we have proved that point L bisects side *BC*.

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