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# Two circles touch each other externally at point p . Apc and bpd are st. Lines. Show that triangle pab and pcd are similar and ab is parallel to cd

Arun
25763 Points
3 years ago
Dear Vaibhav

The angle between  the tangent and any chord at the point of contact is equal to the angle subtended by the arc on the alternate segmenta of the arc.
Let T P T' be the tangent.
The angle T P A = angle A B D ( through the point of contact P), by reason of above theorem.
Similarly, angle T' P C = angle  C D B.
But angle T P A = angle T' P C as they are opposite angle.

Therefore, angle A B D = angle  C D B. So the alternate angles are equal, BD being the transversal.

This implies  AB || CD.

Regards