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old problem

Source: 12th Austrian-Polish MO 1989 problem 8

April 14, 2005
geometrycircumcircleinradiusincenterEulerinequalitiesarea of a triangle

Problem Statement

ABCABC is an acute-angled triangle and PP a point inside or on the boundary. The feet of the perpendiculars from PP to BC,CA,ABBC, CA, AB are A′,B′,C′A', B', C' respectively. Show that if ABCABC is equilateral, then AC′+BA′+CB′PA′+PB′+PC′\frac{AC'+BA'+CB'}{PA'+PB'+PC'} is the same for all positions of PP, but that for any other triangle it is not.
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