Prove that the tangents at the extremities of any chord make equal angles with the chord.

Explanation

GIVEN `AB` is any chord of a circle with centre `O`. Tangents at the extremitites `A` and `B` of this chord meet at an external point `P`. Chord `AB` intersects the line segment `OP` at `M`.
TO PROVE `/_MAP=/_MBP`.
PROOF In `DeltaMAP` and `DeltaMBP`, we have
`PA=PB` [`:.` tangents from an external point are equal]
`MP=MP` [common]
`/_MPA=/_MPB` [ `:' /_OPA=/_OPB` since tangents from an external point are equally inclined to the line segment joining the point to the centre]
`:.DeltaMAP~=DeltaMBP` [by SAS-congruence].
And so, `/_MAP=/_MBP` [cpct]