Line joining circumcenter, incenter is perpendicular to a side for?

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!!

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