MathDB
angle MBC is equal to angle FC iff ABC is equilateral

Source: Moldova TST 2005 for JBMO, day 2, problem 1

April 11, 2005
trigonometrygeometrycircumcircleangle bisectortrig identitiesLaw of Sinesperpendicular bisector

Problem Statement

Let ABCABC be an acute-angled triangle, and let FF be the foot of its altitude from the vertex CC. Let MM be the midpoint of the segment CACA. Assume that CF=BMCF=BM. Then the angle MBCMBC is equal to angle FCAFCA if and only if the triangle ABCABC is equilateral.