Don't know for sure if this example qualifies, but it certainly is a hard problem which becomes trivial from the right point of view. (I learned this from Martin Gardner, proper credits might be researched if necessary).

Problem: three circles in the plane, no two with the same radius, pairwise disjoint. For each pair of circles, there are four straight lines tangent to both; take the two which leave both circles on the same side; they intersect at a point. Repeat the construction for each pair of circles. We get three points: prove that they are collinear.

You may want to think a little about the problem; can be solved both by plane or analytic geometry, with some effort. Not too difficult, but not a one-liner.

Now consider the following solution: add a dimension. You have three spheres, and if you section them through their centers with a plane you get the original three circles. Consider the cones determined by each couple of spheres; the section is the couple of tangent lines seen above, and the tips of the cones are the three points in the problem. Now take two planes touching the three spheres from above and from below....