Prove that the median in a triangle is not less than the height drawn from the same vertex.

Consider a triangle ABC, with a median AK and an altitude AH. Height perpendicular to the side to which it is dropped.
Consider the triangle, which was formed by the height, the median and the side to which they are lowered: AHK, <AHK = 90 °, the triangle is right-angled, the sides that include the point of right angle AH and NK are its legs, and AK is the hypotenuse, for example AK < AH, then by the Pythagorean theorem:
AK²-AH² = НК <0, but this cannot be, the hypotenuse is always larger than the leg, and the height is always the leg. The median can be equal to the height in an equilateral and isosceles triangle, but not less than the height.
Q.E.D.