In a right-angled triangle ABC (angle B is a straight line), point D was taken on the hypotenuse so that BD = DA.

In a right-angled triangle ABC (angle B is a straight line), point D was taken on the hypotenuse so that BD = DA. prove that BD = DC.

We denote the angle DBA = a
Because BD = DA, then triangle BDA is isosceles and angle DAB = DBA = a.
Because the sum of the angles of triangle ABD is 180, then the angle BDA = 180 – a – a = 180 – 2a
Because angles BDA and BDC are adjacent, then BDC = 180 – BDA = 2a
Because ABC is a right angle, then DBC = 90 – ABD = 90 – a
Because the sum of the angles of the triangle BDC is 180, then the angle BCD = 180 – CBD – BDC = 180 – (90 – a) – 2a = 90 -a
Because angles CBD and BCD are equal, then triangle BCD is equilateral and sides BD and CD are equal.