Point D is chosen in an isosceles triangle ABC with base AB on side CB so that CD

Point D is chosen in an isosceles triangle ABC with base AB on side CB so that CD is equal to AC-AB. Point M is the middle of AD. Prove that the angle of the BMC is obtuse.

Solution:

1) AC = BC = CD + DB

CD = CD + DB – AB

DB – AB = 0

DB = AB

2) Triangle ADB is isosceles, then angle A = angle D.

AM = MD, therefore, BM is the median, and also, by definition, the height. That is, the angle is DМВ = 90 degrees.

3) The angle of CMB is greater than the angle of DMB, therefore CMB> 90 degrees, which was required to be proved.