In Russia, we have a lot of different mathematical competitions for a lot of different ages. At the age between 12 and 13, we have to suggest participants some geometry problems that can use only
- congruent triangles
- isosceles triangle
- parallel lines and a sum of the angles of a triangle
- the median in a right triangle
- a right triangle with angle $30^\circ$
- triangle inequality
At the age of 14, we can add
- parallelogram properties
- trapezoid properties
- intercept theorem
- Pythagoras's theorem
- similar triangles
- area
- standard locuses of points (mid perpendicular, bisector)
Sometimes we can add Ceva's and Menelaus's theorems (as a logical continuation of intercept theorem and area), Carnot's theorem. Also, we can add some interesting ideas about concurrent lines: incenters, excenters, orthocenters.
The main restriction is "no cyclic quadrilaterals."
I tried to find some books in English and Romanian that use only these ideas. Or some mathematical competitions.
The only book I found is
- M.N. Aref, William Wernick "Problems & Solutions in Euclidean geometry."
All other books don't contain any problems (only exercises) on these topics or start with cyclic quadrilaterals.
Can you recommend me any books (for example, textbooks), competitions, magazines that contain plenty of such problems? (Practically any competition sometimes have such problems, but very seldom)