Almost every theorem
(Edited after Willie's comment)
Many theorems about a triangle can be proved easily using the fact that all triangles are linearly equivalent. In addition to Morley's theorem, as cited by David Lehavi, other examples include One example is that the angle bisectors intersect in a single point and the segments that join each vertex to the midpoint of the opposite side intersect in a single point.
(One of these This actually appeared on a written qualifying exam I took as a graduate student, and I did in fact use the linear algebra approach, since there was no chance I could recall the Euclidean geometric proof.)