Here’s a way to find tangents to an ellipse from a point outside it, say E. Use E to draw any two chords CD and FG. Now lines CF and DG will meet at H, and CG and DF will meet at J. The line HJ intersects the ellipse at A and B, and EA and EB are the tangents we sought.
In 2001 David Bloom of Brooklyn College wrote, “I owe the above to a course I took in 1958, taught by O. Zariski. The result seemed so beautiful that I’ve never forgotten it.”
(“Miscellanea,” College Mathematics Journal 32:4 [September 2001], 317-318.)