MathDB

5

Part of 2015 Sharygin Geometry Olympiad

Problems(3)

moving two triangles with 1 angle a, to construct angle a/2

Source: Sharygin Geometry Olympiad 2015 Final 8.5

8/1/2018
Two equal hard triangles are given. One of their angles is equal to α \alpha (these angles are marked). Dispose these triangles on the plane in such a way that the angle formed by some three vertices would be equal to α/2 \alpha / 2. (No instruments are allowed, even a pencil.)
(E. Bakayev, A. Zaslavsky)
geometrybisectionangle bisector
Right-angled triangle geometry

Source: Sharygin geometry olympiad 2015, grade 10, Final Round, Problem 5

7/17/2018
Let BMBM be a median of right-angled nonisosceles triangle ABCABC (∠B=90\angle B = 90), and HaH_a, HcH_c be the orthocenters of triangles ABMABM, CBMCBM respectively. Lines AHcAH_c and CHaCH_a meet at point KK. Prove that ∠MBK=90\angle MBK = 90.
geometry
concurrent lines with the line passing through 2 midpoints

Source: Sharygin Geometry Olympiad 2015 Final 9.5

8/1/2018
Let BMBM be a median of nonisosceles right-angled triangle ABCABC (∠B=90o\angle B = 90^o), and Ha,HcHa, Hc be the orthocenters of triangles ABM,CBMABM, CBM respectively. Prove that lines AHcAH_c and CHaCH_a meet on the medial line of triangle ABCABC.
(D. Svhetsov)
geometryconcurrencyconcurrentmidpoint