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It is known that \angle BAC is the smallest angle in the triangle ABC. The points B and C divide the circumcircle of the triangle into two arcs. Let U be an interior point of the arc between B and C which does not contain A. The perpendicular bisectors of AB and AC meet the line AU at V and W, respectively. The lines BV and CW meet at T.

Show that AU = TB + TC.


Alternative formulation:

Four different points A,B,C,D are chosen on a circle \Gamma such that the triangle BCD is not right-angled. Prove that:

(a) The perpendicular bisectors of AB and AC meet the line AD at certain points W and V, respectively, and that the lines CV and BW meet at a certain point T.

(b) The length of one of the line segments AD, BT, and CT is the sum of the lengths of the other two.

Slični zadaci

#NaslovOznakeRj.KvalitetaTežina
1982IMO Shortlist 1997 problem 260
1975IMO Shortlist 1997 problem 192
1969IMO Shortlist 1997 problem 132
1967IMO Shortlist 1997 problem 111
1739IMO Shortlist 1989 problem 10
1705IMO Shortlist 1987 problem 210