AB perpendicular to DB and DC perpendicular to DB, DE = BE. Prove that distance AB is equal to the length of the segment DC.

Since AB is perpendicular to DВ, and DС is perpendicular to ВD, the triangles СDE and ABE are rectangular.
In right-angled triangles СDE and ABE, the angle СED = ВED as vertical angles at the intersection of straight lines ВD and AC, side DE = BE by condition.
Then the right-angled triangles СDE and ABE are equal in the leg and the acute angle adjacent to it, the second sign of equality of right-angled triangles, and then AB = DС, which was required to prove.