Review question

# If these centres lie on a line, why is the triangle isosceles? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R8917

## Suggestion

Prove that the internal bisector of an angle of a triangle divides the opposite side in the ratio of the lengths of the containing sides.

Can we draw a stetch that includes everything we know? Can we somehow apply the sine or cosine rule to the problem?

In a triangle $ABC$, $I$ is the incentre, $H$ the orthocentre, and $O$ the circumcentre. Prove that $AI$ bisects $\angle OAH$.

Can we show these two angles are equal?

Hence, or otherwise, prove that if $O$, $H$ and $I$ are collinear the triangle must be isosceles.

Why are the two lengths labelled $r$ equal here? Can we use our results from the first and second parts?