« Vrati se
Let ABC be a triangle with circumcentre O. The points P and Q are interior points of the sides CA and AB respectively. Let K,L and M be the midpoints of the segments BP,CQ and PQ. respectively, and let \Gamma be the circle passing through K,L and M. Suppose that the line PQ is tangent to the circle \Gamma. Prove that OP = OQ.

Proposed by Sergei Berlov, Russia

Slični zadaci

Let P be a point inside a triangle ABC such that
\angle APB - \angle ACB = \angle APC - \angle ABC.
Let D, E be the incenters of triangles APB, APC, respectively. Show that the lines AP, BD, CE meet at a point.
The circle S has centre O, and BC is a diameter of S. Let A be a point of S such that \angle AOB<120{{}^\circ}. Let D be the midpoint of the arc AB which does not contain C. The line through O parallel to DA meets the line AC at I. The perpendicular bisector of OA meets S at E and at F. Prove that I is the incentre of the triangle CEF.
1. Let ABC be an acute-angled triangle with AB\neq AC. The circle with diameter BC intersects the sides AB and AC at M and N respectively. Denote by O the midpoint of the side BC. The bisectors of the angles \angle BAC and \angle MON intersect at R. Prove that the circumcircles of the triangles BMR and CNR have a common point lying on the side BC.
In triangle ABC the bisector of angle BCA intersects the circumcircle again at R, the perpendicular bisector of BC at P, and the perpendicular bisector of AC at Q. The midpoint of BC is K and the midpoint of AC is L. Prove that the triangles RPK and RQL have the same area.

Author: Marek Pechal, Czech Republic
Let H be the orthocenter of an acute-angled triangle ABC. The circle \Gamma_{A} centered at the midpoint of BC and passing through H intersects the sideline BC at points A_{1} and A_{2}. Similarly, define the points B_{1}, B_{2}, C_{1} and C_{2}.

Prove that six points A_{1} , A_{2}, B_{1}, B_{2}, C_{1} and C_{2} are concyclic.

Author: Andrey Gavrilyuk, Russia
Let ABC be a triangle with AB = AC . The angle bisectors of \angle C AB and \angle AB C meet the sides B C and C A at D and E , respectively. Let K be the incentre of triangle ADC. Suppose that \angle B E K = 45^\circ . Find all possible values of \angle C AB .

Jan Vonk, Belgium, Peter Vandendriessche, Belgium and Hojoo Lee, Korea