Find sufficient conditions in which the triangle $ABC$ is located inside the domain traced by the triangle $DEF$.

Question

Let us consider two triangles $ABC$ and $DEF$ in the plane.

My question is: Find sufficient conditions in which the triangle $ABC$ is located inside the domain traced by the triangle $DEF$.

Answer 1

Let $X$ be a point in the plane of $\Delta DEF$.

Thus, if $$\measuredangle DXE+\measuredangle EXF+\measuredangle DXF=360^{\circ}$$ then $X$ is placed inside the $\Delta DEF$.

If it's not so then $X$ is not placed inside the triangle.

You can check it for any vertex of $\Delta ABC$.

Another way.

If for all point $Y$ of the plane the ray $XY$ intersects some side of the $\Delta DEF$ then $X$ is placed inside the $\Delta DEF$.

If there is ray $XY$, which does not intersect sides of $\Delta DEF$ then $X$ is not placed inside the $\Delta DEF$.