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Belarus geometry
Problem
Point is marked on the side of a triangle . The incircle of the triangle meets the segment at points and . Is it possible that the equalities hold if is
a) the median? b) the bisector? c) the altitude? d) the segment joining vertex with the point of tangency of the excircle of the triangle with ?
a) the median? b) the bisector? c) the altitude? d) the segment joining vertex with the point of tangency of the excircle of the triangle with ?
Solution
a) yes, it is possible; this is valid for the triangles with , , , where is any positive real number;
b) it is impossible;
c) it is impossible;
d) yes, it is possible; this is valid for the triangles with , , , where is any positive real number.
b) it is impossible;
c) it is impossible;
d) yes, it is possible; this is valid for the triangles with , , , where is any positive real number.
Final answer
a) Yes; for triangles with AC = 5x, AB = 10x, BC = 13x. b) No; impossible. c) No; impossible. d) Yes; for triangles with AC = 3x, AB = 4x, BC = 5x.
Techniques
Triangle centers: centroid, incenter, circumcenter, orthocenter, Euler line, nine-point circleTangentsHomothety