ИСТИНА

Euclidean geometry is a difficult subject for both school students and professional mathematicians as both groups struggle to understand the nature of various tricks that differ from problem to problem. The second group at least can declare every geometry problem trivial by means of coordinatization and computer algebra. Such an approach may close a problem by academic standards, but it does not necessarily mean understanding. Putting problem in a nontrivial context makes it easier and more accessible for a mathematician. In this talk I will consider three problems of Euclidean Geometry: 1. A convex quadrilateral with sides $a,b,c,d$ and diagonals $e$, $f$ can be inscribed in a circle if and only if $abe-bcf+cde-daf=0$ (each term corresponding to the product of sides of a triangle). 2. The distance $IO$ between the incenter $I$ and the circumcenter $O$ of a triangle is $\sqrt{R(R-2r)}$, where $R$ and $r$ are the circumradius and inradius, respectively. 3. The distance $IN$ between the incenter $I$ and the nine-point center $N$ (the circumcenter of the median triangle) is $(R-2r)/2$. All the three have an interesting history, with such names as Ptolemy, Ramanujan, Euler, Poncelet, Cayley, Feuerbach involved. Despite of availability of ``trivial'' solutions, attempts to understand these problem lead to topics ranging from approximation theory for multivariable polynomials to elliptic functions to a less known ``augmented space of conics'', through which I am trying to learn algebraic geometry.