To explore Varignon's Theorem click in the canvas below. This will construct a quadrilateral, where the mid points of each edge are marked and form a nother quadrilateral, marked in yellow. The theorem states, that this shape is always a parallelogram. After the construction you can alter the shape of the quadrilateral by dragging its vertices around the canvas.
Activating the triangles option in the shows two purple triangles, that can be scaled with the slider. Observe that at maximum scale the triangles are formed by two edges of the quadrilateral and the diagonal. The two triangles share an edge! At any scale the scaled triangles are congruent to the triangle formed by a vertex of the quadrilateral and the middle points of the edges in at that vertex, hence the edge of the yellow parallelogram. During that scaling the edge, that doesn't lie on edges of the quadrilateral stays parallel to the edge connecting the middlepoints. This concludes, that the opposite edges of the yellow shape must be parallel. Repeating the same process for the other two edges, completes the proof.
Note that, this is clearly not a rigorous proof, but it is a visualization of an actual proof that can be performed in mathematical terms.