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