prove that only two equal tangents can be drawn on a circle from a point given
outside the circle
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Let the center of the circle be O and a point outside the circle be P. Suppose T is a a point of tangency so that the line PT is tangent to the circle. Then the radius OT is perpendicular to the tangent line. The triangle OPT is a right triangle, so T lies on the circle whose diameter is OP.
Thus every point of tangency lies in the given circle as well as the circle whose diameter is OP. Two circles intersect in at most two points. Therefore at most two tangents can be drawn from an external point of a circle.
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