1. **State the problem:** We have a triangle ABC with a circle centered at point O inside it. Angle at vertex A is given as 46°. The circle passes through points B and C, so B and C lie on the circle with center O. 2. **Understanding the geometry:** Since the circle is inside the triangle and passes through points B and C, segment BC is a chord of the circle. Point O is the center of the circle. 3. **Key observation:** Angle A is circumscribed about circle O, meaning angle A intercepts arc BC on the circle. The angle at the center O that subtends the same chord BC is angle BOC. 4. **Relating inscribed angle to central angle:** The measure of an inscribed angle (angle A, which subtends chord BC) is half the measure of the central angle that subtends the same chord. That is: $$\angle A = \frac{1}{2} \angle BOC$$ 5. **Calculate angle O (central angle):** Given $$\angle A = 46^\circ$$, then $$\angle BOC = 2 \times 46^\circ = 92^\circ$$ **Final answer:** $$\boxed{\angle O = 92^\circ}$$