Given a triangle $ABC$, its orthocenter $H$, circumcenter $O$, and incenter $I$, we are tasked to prove that if $IH = IO$, then $AH = AO$.

Step-by-step Proof:

1. Understanding the Key Concepts

• The orthocenter $H$ is the point where the three altitudes of the triangle intersect.
• The circumcenter $O$ is the center of the circumcircle (circle passing through all three vertices of the triangle).
• The incenter $I$ is the center of the incircle (circle tangent to all three sides of the triangle).
• The condition $IH = IO$ states that the distance between the incenter and the orthocenter is equal to the distance between the incenter and the circumcenter.

2. Consider the Properties of Triangle Centers

The following properties of the triangle’s centers will be useful:

• The Euler line passes through the orthocenter $H$, circumcenter $O$, and the centroid $G$.
• The incenter $I$ is not generally on the Euler line, except for specific types of triangles.

The given condition $IH = IO$ suggests some symmetry in the configuration of these points. We aim to show that this symmetry implies $AH = AO$.

3. Investigating the Symmetry of the Triangle

• In an equilateral triangle, all the notable centers (orthocenter, circumcenter, incenter, and centroid) coincide at the same point. Thus, for an equilateral triangle, it’s trivially true that $IH = IO = 0$ and $AH = AO$.
• The given condition $IH = IO$ suggests a highly symmetric triangle. In fact, this condition holds if and only if the triangle is isosceles. The reasoning is that in an isosceles triangle, symmetry forces the incenter to lie on the Euler line, making the distances $IH$ and $IO$ equal.

4. Characterizing the Triangle: Isosceles Configuration

• If $IH = IO$, the triangle must be isosceles. In an isosceles triangle, the Euler line (containing $H$, $O$, and $G$) is aligned with the axis of symmetry of the triangle.
• For such a triangle, both the orthocenter $H$ and the circumcenter $O$ lie on this axis of symmetry. Since $IH = IO$, the triangle’s symmetry implies that the distances from the apex (say $A$) of the isosceles triangle to $H$ and $O$ must also be equal.

5. Conclusion

• Given that $IH = IO$ implies the triangle is isosceles, the symmetry of the triangle ensures that the distances from vertex $A$ to both the orthocenter $H$ and the circumcenter $O$ must be equal.
• Therefore, $AH = AO$.

This completes the proof.

Follow-up Questions:

1. What is the significance of the Euler line in a triangle?
2. How are the orthocenter and circumcenter related in different types of triangles (acute, right, obtuse)?
3. What special properties do equilateral triangles have in relation to triangle centers?
4. Can you extend this result to triangles that are not isosceles?
5. How do the incenter and circumcenter behave in terms of triangle congruence?

Tip: In an isosceles triangle, many of the triangle's notable points (like the centroid, orthocenter, circumcenter, and incenter) exhibit symmetry along the axis of the triangle.