I like the lovely theorem in 19th century Euclidean Geometry as follows.
Let ABC be a triangle. let D,E,F be points on BC,CA,AB respectively. Then the circumcircles of AFE, BDF, CDE meet at a point.
I like this because the proof uses the property of the angles of cyclic quadrilaterals, and its converse. Also if one wants to convince students of the necessity of proof, then one should start with a result which is surprising.
It is a good thing that this situation can we worked on for more implications. Let P,Q,R be the centres of the three circles just given. Then the triangle PQR is similar to the triangle ABC.
For all these reasons I think it is a pity that some of Euclidean Geometry is not in University courses, or often school courses, in order to acquaint students with something important in our mathematical heritage. Should a student get a degree in maths without knowing why the angle in a semicircle is a right angle?