2010.6

Part of Ukrainian TYM Qualifying - geometry

Problems(1)

geometric and harmonic mean of distances of sides of triangle, max values

Source: 2010 XIII All-Ukrainian Tournament of Young Mathematicians, Qualifying p6

Find inside the triangle

ABC

G

H

for which, respectively, the geometric mean and the harmonic mean of the distances to the sides of the triangle acquire maximum values. In which cases is the segment

GH

parallel to one of the sides of the triangle? Find the length of such a segment

GH

geometrymaxgeometric inequalityUkrainian TYM