MathDB

18

Part of 2005 Sharygin Geometry Olympiad

Problems(1)

concurrency of midpoints of symmetric points wrt 3 lines forming triangle

Source: Sharygin 2005 X,XI CR 18

8/19/2019
On the plane are three straight lines ℓ1,ℓ2,ℓ3\ell_1, \ell_2,\ell_3, forming a triangle, and the point OO is marked, the center of the circumscribed circle of this triangle. For an arbitrary point X of the plane, we denote by XiX_i the point symmetric to the point X with respect to the line ℓi,i=1,2,3\ell_i, i = 1,2,3. a) Prove that for an arbitrary point MM the straight lines connecting the midpoints of the segments O1O2O_1O_2 and M1M2,O2O3M_1M_2, O_2O_3 and M2M3,O3O1M_2M_3, O_3O_1 and M3M1M_3M_1 intersect at one point, b) where can this intersection point lie?
concurrencyconcurrentsymmetrylinesCircumcentergeometry