Sclene Triangle Property

An acute-angled triangle has unequal sides. Show that the line through the circumcenter and incenter intersects the longest side and the shortest side.

 

Solution

Let the vertices be A, B, C. Assume AB > BC > CA. Let the circumcenter be O and the incenter be I. Extend the perpendicular from O to BC to meet the circle at M, so that M is the midpoint of the arc BC. Then AM is the angle bisector of A, so passes through I. Since AB > AC, O must lie on the same side of the line AI as B. Similarly, since BA > BC, O must lie on the same side of the line BI as A. So the ray IO lies between the rays IA and IB and hence intersects AB, the longest side.

Similarly, since CB > CA, O must lie on the same side of CI as B. Thus O lies between the rays CI and IB. Hence the ray OI must lie between the rays IC and BI and hence intersect AC, the shortest side.