There is a well known theorem in plane Euclidean geometry as follows, and I would like to know only the name by which it is known.
Theorem: Let $ABC$ be a triangle. Choose points $D,E,F$ on the sides $BC, CA, AB$ respectively. Then the circumcircles of the triangles $AEF, BFD,CDE$ meet at a point.
I like this for first year students since it introduces the idea of proof for something which is a surprise, rather than something boring, and also it uses the theorem on angles in a cyclic quadrilateral, and its converse. Further, it can be developed further, such as proving the circumcentres of these circles form a triangle similar to $ABC$.