In triangle ABC BM is the median and BH is the height. It is known that AC = 79 and BC = BM. Find AH

Since BM is the median, it means AM = MC = AC / 2 = 79/2 = 39.5
Consider the MBC triangle.
Because BC = BM (by the condition of the problem), then this triangle is isosceles, BH is the height of this triangle. According to the third property of the isosceles triangle, MH = HC = MC / 2 = 39.5 / 2 = 19.75
The desired AH = AC-HC = 79-19.75 = 59.25
Answer: AH = 59.25