In the diagram shown, O and P are centres, LMN is a common tangent and MT a common chord.

i) Prove that OLMT and PNMT are cyclic quadrilaterals.

ii) Prove that triangle LMT is similar to triangle TNP.

For ii), the solution gave:

In triangles LTM and TNP,

Angle LMT = Angle TPN (exterior angle of cyclic quad. equal to interior remote angle)

PT = PN (equal radii)

LM = MT (tangents from external point equal)

Therefore, triangle LMT is similar to triangle TNP (corresponding sides about equal angles)

Could someone explain me how PT = PN and LM = MT works to prove the triangles are similar?

Thanx a lot!