in circle, TP is a tangent and PAB is a secant to the circle. If the bisector of angleATB intersects AB at M, show that

1 angle PMT= anglePTM

2 PT=PM


As angle PTA = angle PTB ( tangent -angle theorem) (1)
And Angle BTM = angle MTA  (given) (2)
And angle PMT = angle PBT + angle MTB
So angle PMT = angle PTA + angle MTA (using (1) and (2) )
And angle PTA + angle MTA = PTM
So angle PMT = angle PTM ( hence proved)

In triangle PMT
We have angle PMT = angle PTM
So PM = PT (isosceles  triangle)


 

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