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# RMO 2008 Solution of Problem 1 Cyclic Pentagon

### Problem

Let ABC be an acute-angled triangle, let D, F be the mid-points of BC, AB respectively. Let the perpendicular from F to AC and the perpendicular at B to BC meet in N. Prove that ND is equal to circum-radius of ABC.

### Theorems and tools

The discussion uses the following Theorems:

Midpoint Theorem: The line segment joining midpoint of two sides of a triangle is half of and parallel to the third side.

Cyclic Quadrilateral: Opposite angles of a quadrilateral add up to $180^o$