Question

If S,I are the circumcenter and incenter of a triangle. Then SI is perpendicular to a side of triangle for:

• Any triangle
• Any right angled triangle
• Any isosceles triangle
• None of these

Solution:

Consider a triangle ABC and let SI be perpendicular to BC.

S lies on the perpendicular bisector of the side.

I lies on the angle bisector of the angles of a triangle.

If SI is the perpendicular to side BC. I also lies on the perpendicular bisector of BC because S lies on the perpendicular of BC

Which implies angle B/2=C/2 so ABC is an isosceles

Alternatively:

Some may remember that for an iscoceles triangle orthocenter, incenter and circumcenter lie on the same line. So for an isosceles triangle ABC SI is perpendicular to BC