Inside the undeveloped angle A, point D is taken, from which the perpendiculars DB and DC

Inside the undeveloped angle A, point D is taken, from which the perpendiculars DB and DC are drawn to the sides of the angle. Angle ADB = angle ADC. Prove that the ray AD is the bisector of angle A.

Since, by condition, DB and DC are perpendiculars to AB AC, the triangles ABD and ACD are rectangular.

In right-angled triangles ABD and ACD, the angle ADB = ADC by condition, and the hypotenuse AD is common in construction, then right-angled triangles ABD and ACD are equal in the third sign of equality of right-angled triangles, in hypotenuse and acute angle, which means the angle BAD = angle CAD, and therefore , AD is the bisector of the angle BAC, as required.