Problem

The key is recognizing that point $O$ is the first isogonic center (or first Brocard point), where equal angles are "swept" from each vertex to the opposite side. Rather than chasing angles directly, use the area method: express the total area of $\triangle ABC$ as a sum of three smaller triangles formed by $O$, then relate those areas to the cotangents of the angles. The constraint that all three angles equal $w$ creates a special relationship that forces $\cot w$ to equal the sum of the triangle's angle cotangents.

The key is recognizing that point $O$ is the first isogonic center (or first Brocard point), where equal angles are "swept" from each vertex to the opposite side. Rather than chasing angles directly, use the area method: express the total area of $\triangle ABC$ as a sum of three smaller triangles formed by $O$, then relate those areas to the cotangents of the angles. The constraint that all three angles equal $w$ creates a special relationship that forces $\cot w$ to equal the sum of the triangle's angle cotangents.