MathDB

10.7

Part of III Soros Olympiad 1996 - 97 (Russia)

Problems(2)

equilateral ABC, B_1C_1 = AM, C_1A_1 = BM, A_1B_1 = CM,

Source: III Soros Olympiad 1996-97 R1 10.7 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

5/29/2024
An arbitrary point MM is taken inside a regular triangle ABCABC. Prove, that on sides ABAB, BCBC and CACA one can choose points C1C_1, A1A_1 and B1B_1, respectively, so that B1C1=AMB_1C_1 = AM, C1A1=BMC_1A_1 = BM, A1B1=CMA_1B_1 = CM. Find BABA if AB1=aAB_1= a, AC1=bAC_1 = b, a>ba>b.
geometryEquilateral
never too late for another locus

Source: III Soros Olympiad 1996-97 R3 10.7 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

5/31/2024
Let AA be a fixed point on a circle, BB andC C be arbitrary points on the circle different from AA and at different distances. The bisector of the angle ∠BAC\angle BAC intersects the chord BCBC and the circle at points KK and PP, DD is the projection of AA onto the straight line BCBC. A circle passing through points KK, PP and DD intersects the straight line ADAD for the second time at point MM. Find the locus of points MM.
geometryLocus