I would like to prove or find a counterexample for the following theorem:
For any $\triangle ABC, \odot P_1, \odot P_2$ such that the three lines $AB, AC, BC$ are each contained in the union of the two circles $P_1 \cup P_2$, the entire interior of $\triangle ABC$ is contained in $P_1 \cup P_2$.
If the theorem holds in Euclidiean geometry, does it also hold in spherical geometry and/or in isotropic Lobachevski geometry (constant negative curvature)?
Euclidean case: Let $X$ be an interior point of $\Delta ABC$ not covered by $P_1\cup P_2$. Then the line $\ell$ through $X$ orthogonal to the connection of the centers intersects $P_1\cup P_2$ in a (convex) line segment. Hence this line segment cannot contain both points where $\ell$ leaves the triangle.