Let $ABC$ be a triangle with circumcenter $O$ and let $X$, $Y$, $Z$ be the midpoints of arcs $BAC$, $ABC$, $ACB$ on its circumcircle. Let $G$ and $I$ denote the centroid of $\triangle XYZ$ and the incenter of $\triangle ABC$.

Given that $AB = 13$, $BC = 14$, $CA = 15$, and $\frac {GO}{GI} = \frac mn$ for relatively prime positive integers $m$ and $n$, compute $100m+n$.

[i]Proposed by Evan Chen[/i]
1. **Identify Key Points and Definitions:**
   - Let \( D, E, F \) be the midpoints of the minor arcs \( BC, CA, AB \) respectively on the circumcircle of \( \triangle ABC \).
   - Let \( G' \) be the reflection of \( G \) over \( O \).
   - Note that \( X, Y, Z \) are the midpoints of the arcs \( BAC, ABC, ACB \) respectively.

2. **Homothety and Symmetry:**
   - Consider a negative homothety centered at \( O \) with a scale factor of \(-1\). This homothety maps \( \triangle XYZ \) to \( \triangle DEF \).
   - Since \( D, E, F \) are the midpoints of the minor arcs, \( AD \perp EF \), \( BE \perp DF \), and \( CF \perp DE \).

3. **Orthocenter and Euler Line:**
   - By symmetry and the properties of the circumcircle, the incenter \( I \) of \( \triangle ABC \) is the orthocenter of \( \triangle DEF \).
   - The Euler line of \( \triangle DEF \) contains \( O \) (the circumcenter), \( G' \) (the centroid), and \( I \) (the orthocenter).

4. **Ratio of Distances:**
   - On the Euler line of \( \triangle DEF \), the distance from the orthocenter to the centroid is one-third the distance from the orthocenter to the circumcenter. Thus, \( \frac{OG'}{OI} = \frac{1}{3} \).
   - Since \( G' \) is the reflection of \( G \) over \( O \), the distances \( OG \) and \( OG' \) are equal, and \( G \) and \( G' \) are collinear with \( I \) and \( O \).

5. **Final Ratio Calculation:**
   - Given that \( I, G', O, G \) are collinear in that order, we have:
     \[
     \frac{GO}{GI} = \frac{OG'}{OI + OG'} = \frac{\frac{1}{3}}{1 + \frac{1}{3}} = \frac{\frac{1}{3}}{\frac{4}{3}} = \frac{1}{4}
     \]

6. **Determine \( m \) and \( n \):**
   - From the ratio \( \frac{GO}{GI} = \frac{1}{4} \), we identify \( m = 1 \) and \( n = 4 \).
   - Therefore, \( 100m + n = 100 \times 1 + 4 = 104 \).

The final answer is \( \boxed{104} \).