Some Combinational Problems in Geometry

In this short survey I mainly discuss some recent problems which occupied me and my colleagues and collaborators for the last few years . I will not give proofs but will either give references to the original papers or to the other survey papers . I hope I will be able to convince the reader that the subject is "alive and well" with many interesting, challenging and not hopeless problems . Development in geo__ry of the kind whic ;a Eucli~'• su ely recognize as gecr:etr, wars perhaps Oct very significant one very striking theorem was proved about 75 years ago by Morley which states that if one trisects the angles a, s, y of a triangle the pairs of trisectors of the angles meet in an equilateral triangle . Surely a striking and unexpected result of great beauty . In fact because the trisection can not be carried out by ruler and compass it is not quite sure if Euclid would recognize this as legitimate (if and when I meet him [soon?] I plan to ask him) . On the other hand, many new results have been found on geometric inequalities I won't deal with them in this paper and state only one of them, the so called Erd6sMordell inequality (which was one of my first conjectures I conjectured it in 1932 two years later Mordell found the first proof) . Let A, B, C be any triangle, 0 a point in its interior, OX is perpendicular