In a convex quadrilateral ABCD, the diagonal AC is the bisector of angles A and C

| September 26, 2021 | education

In a convex quadrilateral ABCD, the diagonal AC is the bisector of angles A and C, the diagonal BD is the bisector of angles B and D. Prove that all sides of the quadrilateral ABCD are equal.

AC is the bisector of angles A and C.

Consider a triangle ABC, in which angle BAC = A / 2, angle BCA = C / 2, angle ABC = B.

Since the sum of the interior angles of a triangle is 180, then:

A / 2 + B + C / 2 = 180.

Similarly, in triangle ACD, the sum of the interior angles is:

A / 2 + D + C / 2 = 180.

Subtract the other from one equality and get:

B – D = 0.

Angle B = D.

Similarly, the angle A = C.

Then the triangles ABC and ACD are equal in three angles, as well as isosceles, since the angles at the base of the AC are equal, and therefore AB = BC = CD = AD, which was required to prove.



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