The segments AB and CD are the diameters of the circle. Prove that:

The segments AB and CD are the diameters of the circle. Prove that: a) the chords BD and AC are equal; b) chords AD and BC; c) angle BAD = BCD.

Consider the triangles AOС and ВOD. In the triangles AOC and ВOD OA = OВ = OC = OD = R.

Angle AOС = ВOD as vertical angles at the intersection of diameters AB and СD. Then the triangles AOС and ВOD are equal on two sides and the angle between them, the first sign of the equality of the triangles. Then the chord ВD = AC, as required.

Similarly, the triangle AOD is equal to the triangle BOС, and then AD = BC.

The inscribed ВAD angle rests on the ВD arc as well as the inscribed BCD angle. The inscribed angles resting on the same arc are equal, as required.