In the quadrilateral ABCD, the bisector of angle A and the bisector of angle B are drawn.

| May 4, 2021 | education

In the quadrilateral ABCD, the bisector of angle A and the bisector of angle B are drawn. The bisector of angle A intersects the side BC at point M, and the bisector of angle B intersects the side AD at point N. It is known that MCDN is a parallelogram. Prove that ABCD is a parallelogram.

Since, by condition, MCDH is a parallelogram, then MC is parallel to DH, and CD is parallel to MH.

The segment CM lies on the segment C, and DH is on AD, then BC is parallel to AD.

In the quadrilateral ABMH AM and BH there are bisectors and diagonals of the quadrilateral.

Triangles ABM and AHM are equal, since the side AM is common, and the angles BAM, BMA, AMH and MAH are equal, since AM and BH are bisectors and intersect two parallel straight lines BC and AD. Then AB = MH, BM = AH, and therefore ABMH is a parallelogram, and AB is parallel to MH, and then AB is parallel to CD. Consequently, the opposite sides of the ABCD quadrangle are parallel, bargaining ABCD is a parallelogram, which was required to be proved.



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