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5

Part of 2005 Junior Balkan Team Selection Tests - Moldova

Problems(1)

angle MBC is equal to angle FC iff ABC is equilateral

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

4/11/2005
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.
trigonometrygeometrycircumcircleangle bisectortrig identitiesLaw of Sinesperpendicular bisector