Question

If PAB is a secant to a circle and PT is a tangent to the same circle then prove that:
(PT)square=(PA)*(PB).

Answer 1

Please draw the figure as you read below.

Join point T to point A and also to point B.
Now you observe two triangles 1. triangle PTA  2. triangle PBT

Also remember:Angle between a chord and a tangent at the point of contact 
= angle,subtended by the same chord, at any point in the opposite segment.

TA is secant. TP is a tangent.
Therefore angle PTA (angle between tangent and chord) 
= angle TBA (angle in the opposite segment)             ----------------(1)

angle PTA = angle TBP  ------------------------------------(2)

angle PTA in triangle PTA = angle TBA in triangle PBT [from (1) above] ---(3)
angle TPA in triangle PTA = angle TPB in triangle PBT (common angle) --(4)

From (3) and (4), two angles of triangles PTA and PBT are equal,
hence triangles PTA and PBT are similar.

Therefore PT / PB = PA / PT OR PT² = PAxPB