Question

If triangle ABC is an isoceles with AB=BC , prove that the tangent at A to the circumcircle of triangle ABC is parallel to BC.

Answer 1

Let DAE be tangent at A to the circumcicle of ΔABC.

In ΔABC,

AB = AC  (Given)

∴ ∠ACB = ∠ABC  ...(1)  (Equal sides have equal angles opposite to them)

We know that, If a line touches a circle and from the point of contact, a chord is drawn, then the angles between  the tangent and the chord are respectively equal to the angles in the corresponding alternative segment.

Now, DAE is the tangent and AB is the chord.

∴ ∠DAB = ∠ACB  ...(2)

From (1) and (2), we have

∠ABC = ∠DAB

∴ DE || BC  (If a transversal intersects two lines in such a way that a pair of alternate interior angles are equal, then the two lines are parallel)

Thus, the tangent at A to the circumcircle of  ΔABC is parallel BC.

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