Geometrical inequality, similar to treegoner's one
Source: Moldavian Olympiad
November 6, 2004
inequalitiesgeometrycircumcircleinradiusfunctiontrigonometrygeometric transformation
Problem Statement
Let be a triangle, let be its circumcenter, and let be its orthocenter.
Let be a point on the segment .
Prove that
,
where is the inradius and the circumradius of triangle .
Moderator edit: This is true only if the point lies inside the triangle . (Of course, this is always fulfilled if triangle is acute-angled, since in this case the segment completely lies inside the triangle ; but if triangle is obtuse-angled, then the condition about lying inside the triangle is really necessary.)