11398. Suppose acute triangle ABC has its middle-sized angle at A. Suppose further that the incenter
I is equidistant from the circumcenter O and the orthocenter H. Show that angle A has measure 60 degrees
and that the circumradius of IBC is the same as that of ABC.
Solution by the Fallbrook Epsilon Academy Problem Group, Fallbrook, CA.
Some definitions:
- incenter
- the point where the 3 angle bisectors of the triangle meet
- circumcenter
- the point where the 3 perpendicular bisectors of the triangle meet; also, the center of the
circle (the circumcircle or circumscribed circle) containing the 3 vertices of the triangle
- orthocenter
- the point where the 3 altitudes of the triangle meet
- circumradius
- the radius of the circumcircle; it is equal to
abc/sqrt((a+b+c)(b+c-a)(c+a-b)(a+b-c))
where the lengths of the 3 sides are a,b,c