BD is the height of triangle ABC. BE is the median of this triangle. Prove that BD is less than BE.

Since the segment BD is the height of the triangle ABC, the angle BDC and the angle BDE are equal to 90. Triangle BDE is rectangular, in which the median BE is the hypotenuse of a right triangle.

Since the square of the hypotenuse is equal to: BE ^ 2 = BD ^ 2 + ED ^ 2, and BD ^ 2 = BE ^ 2 – ED ^ 2, then BD ^ 2 <BE ^ 2, and therefore BD <BE, as required prove.