Olympiad problems

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Found problems: 2

II Soros Olympiad 1995 - 96 (Russia), 10.9

The opposite sides of a quadrilateral inscribed in a circle intersect at points $K$ and $L$. Let $F$ be the midpoint of $KL$, $E$ and $G$ be the midpoints of the diagonals of the given quadrilateral. It is known that $FE = a$, $FG = b$. Calculate $KL$ in terms of $a$ and $b.$ (It is known that the points $F$, $E$ and $G$ lie on the same straight line. This is true for any quadrilateral, not necessarily inscribed. The indicated straight line is sometimes called the Newton−Gauss line. This fact can be used without proof in proving the problem, as it is known).

Geometry Mathley 2011-12, 13.2

Tags: excircle , euler circle , euler line , newton line , geometry

In a triangle $ABC$, the nine-point circle $(N)$ is tangent to the incircle $(I)$ and three excircles $(I_a), (I_b), (I_c)$ at the Feuerbach points $F, F_a, F_b, F_c$. Tangents of $(N)$ at $F, F_a, F_b, F_c$ bound a quadrangle $PQRS$. Show that the Euler line of $ABC$ is a Newton line of $PQRS$. Luis González