We are given a circle $ k$ and nonparallel tangents $ t_1,t_2$ at points $ P_1,P_2$ on $ k$, respectively. Lines $ t_1$ and $ t_2$ meet at $ A_0$. For a point $ A_3$ on the smaller arc $ P_1 P_2,$ the tangent $ t_3$ to $ k$ at $ P_3$ meets $ t_1$ at $ A_1$ and $ t_2$ at $ A_2$. How must $ P_3$ be chosen so that the triangle $ A_0 A_1 A_2$ has maximum area?
1. **Given Setup**: We are given a circle \( k \) with nonparallel tangents \( t_1 \) and \( t_2 \) at points \( P_1 \) and \( P_2 \) on \( k \), respectively. The tangents \( t_1 \) and \( t_2 \) intersect at point \( A_0 \). For a point \( P_3 \) on the smaller arc \( P_1P_2 \), the tangent \( t_3 \) to \( k \) at \( P_3 \) meets \( t_1 \) at \( A_1 \) and \( t_2 \) at \( A_2 \).

2. **Objective**: We need to determine how \( P_3 \) should be chosen so that the triangle \( A_0A_1A_2 \) has maximum area.

3. **Tangents and Intersection Points**: 
   - Let \( t_1 \) be the tangent at \( P_1 \) and \( t_2 \) be the tangent at \( P_2 \).
   - The tangents \( t_1 \) and \( t_2 \) intersect at \( A_0 \).
   - The tangent \( t_3 \) at \( P_3 \) intersects \( t_1 \) at \( A_1 \) and \( t_2 \) at \( A_2 \).

4. **Symmetry and Equal Angles**:
   - Since \( P_3 \) is on the smaller arc \( P_1P_2 \), the angles \( \angle P_1P_3P_2 \) and \( \angle P_2P_3P_1 \) are equal.
   - The tangents \( t_1 \) and \( t_2 \) form equal angles with the line segment \( P_1P_2 \).

5. **Area of Triangle \( A_0A_1A_2 \)**:
   - The area of triangle \( A_0A_1A_2 \) can be maximized by maximizing the perpendicular distance from \( A_0 \) to the line segment \( A_1A_2 \).
   - This perpendicular distance is maximized when \( A_1 \) and \( A_2 \) are as far apart as possible, which occurs when \( P_3 \) is the midpoint of the smaller arc \( P_1P_2 \).

6. **Conclusion**:
   - To maximize the area of triangle \( A_0A_1A_2 \), \( P_3 \) must be chosen such that it is the midpoint of the smaller arc \( P_1P_2 \).

The final answer is \( \boxed{ P_3 } \) must be the midpoint of the smaller arc \( P_1P_2 \).