Source — Euclid's Elements, encoded as an rrxiv paper

as centre and

as radius describe a circle

, meeting the perpendicular at

. Join

, meeting the original circle

at

. Then

is the desired tangent. Proof:

and

are congruent by SAS (

common,

both equal to the radius of

,

by construction); hence

, which is right. So

, the radius at the point of contact, and by III.18 (next)

is tangent.