The tangent to the circumcircle of an isosceles triangle \(\triangle ABC\) at \(A\) (where \(AB=AC\)) is parallel to \(BC\). State True/False and justify.
True.
Step 1: Recall the tangent–chord theorem. It says: the angle between a tangent and a chord drawn at the point of contact is equal to the angle made by the chord in the opposite arc of the circle.
Step 2: In \(\triangle ABC\), a tangent is drawn at point \(A\). - The tangent at \(A\) and the chord \(AB\) make an angle. By the theorem, this angle is equal to \(\angle ACB\). - Similarly, the tangent at \(A\) and the chord \(AC\) make another angle. By the theorem, this angle is equal to \(\angle ABC\).
Step 3: Since the triangle is isosceles (\(AB = AC\)), the base angles are equal. So, \(\angle ABC = \angle ACB\).
Step 4: That means the tangent at \(A\) makes equal angles with \(AB\) and \(AC\). But these are exactly the same angles that the base \(BC\) makes with \(AB\) and \(AC\).
Step 5: Therefore, the tangent line at \(A\) is parallel to the base \(BC\).
Final Answer: The statement is True.