MathDB

3

Part of 2011 Moldova Team Selection Test

Problems(2)

equilateral triangles build on sides of a quadrilateral

Source: Moldova TST, day 1, problem 3

3/8/2011
Let ABCDABCD be a quadrilateral and MM the midpoint of the segment ABAB. Outside of the quadrilateral are constructed the equilateral triangles BCEBCE, CDFCDF and DAGDAG. Let PP and NN be the midpoints of the segments GFGF and EFEF. Prove that the triangle MNPMNP is equilateral.
geometrygeometric transformationrotationvectorgeometry unsolved
Bisectors in triangle

Source: Moldova TST, day 2, problem 3

3/8/2011
Let ABCABC be a triangle with ∠BAC=60\angle BAC=60. Let B1B_1 and C1C_1 be the feet of the bisectors from BB and CC. Let A1A_1 be the symmetrical of AA according to line B1C1B_1C_1. Prove that A1,B,CA_1, B, C are colinear.
geometryincentergeometry unsolved