To construct an isosceles triangle having each of the angles at the base double of the remaining one.
Proof
Take a straight line AB and cut it at C so that the rectangle on
AB, BC equals the square on AC (II.11, the golden section).
With centre A and radius AB describe a circle; in it apply chord
BD equal to AC (IV.1). Join AD, CD. Because AB⋅BC=AC2=BD2, BD is tangent to the circle through A, C, D
(III.37); by III.32 the tangent–chord angle equals the alternate
inscribed angle. Tracking the resulting angle relations (with I.5
for the isosceles base angles) gives the required ratio.