The square ADEF is inscribed in a right-angled triangle ABC with a right angle A and legs AB = 2

The square ADEF is inscribed in a right-angled triangle ABC with a right angle A and legs AB = 2, AC = 6. Prove that the triangles BDE and EFC are similar.

Since ADEF is a square, the angle ADE = AFE = 90.

ED is perpendicular to AD, hence perpendicular to AB, then triangle BED is rectangular.

FЕ is perpendicular to AF, hence also perpendicular to AC, then the triangle CEF is rectangular.

FE is parallel to AD as sides of the square, which means that FE is parallel to AB. Then the angle DBE is equal to the angle FEC, as are the corresponding angles at the intersection of parallel straight lines FE and AB secant BC.

Then the right-angled triangles BDE and EFC are similar in acute angle, as required.