Soo, I've been going trough the ukmt handbook "Plane Euclidean Geometry" by A.D. Gardiner and C. J Bradley and I am very stuck on one of the questions - Question 4 of 1g section. It basically asks to prove that if two medians are equal the triangle is isosceles. The problem is that I want to solve it using only the material from chapter 1 (parallel lines, triangle congruencess, Pythagoras theorem , areas of triangles). I can't think of a way to prove it without first proving that medians divide each other in 2:1 ratio, but in any proof of that I'm trying I need to use the midpoint theorem, which as far as I know is proved through similarities which are studied in chapter 2, so I would assume they are not intended to be used yet.. (I could do it with vectors, but probably it is also not the intended method) Anyone has suggestions that don't involve similarities?