Seventh book of series the "School mathematical mugs" sanctified to the classic averages, majority from that were known yet in antiquity, and to applications of their properties at the decision of arithmetic, algebraic and geometrical tasks. The special attention is spared to intercommunication of different averages and establishing intersubject connections between some themes of school courses of algebra and geometry. A book is intended for reading with schoolchildren 5-11 classes. She was entered by developments of ten employments of mathematical group with the examples of different complication, tasks for an independent decision and methodical pointing, taken apart in detail for a teacher.A large list over of additional tasks of different level of difficulty is brought also. For comfort of the use final part of book is done as distributing materials.A book is addressed to the teachers of mathematics and leaders of mathematical groups.