MathDB

6

Part of 2010 Sharygin Geometry Olympiad

Problems(4)

The line AK passes through circumcenter (6)

Source:

10/28/2010
Points MM and NN lie on the side BCBC of the regular triangle ABCABC (MM is between BB and NN), and ∠MAN=30∘.\angle MAN=30^\circ. The circumcircles of triangles AMCAMC and ANBANB meet at a point K.K. Prove that the line AKAK passes through the circumcenter of triangle AMN.AMN.
geometrycircumcircleperpendicular bisectorgeometry unsolved
equal angles inside square ABCD, midpoints of BC and CD related

Source: Sharygin 2010 Final 8.6

10/2/2018
Let E,FE, F be the midpoints of sides BC,CDBC, CD of square ABCDABCD. Lines AEAE and BFBF meet at point PP. Prove that ∠PDA=∠AED\angle PDA = \angle AED.
geometryanglessquaremidpointsAngle Chasinggeometry solvedSharygin Geometry Olympiad
locus of circumcenters of AMK, related to a line passing though vertex B

Source: Sharygin 2010 Final 9.6

10/6/2018
An arbitrary line passing through vertex BB of triangle ABCABC meets side ACAC at point KK and the circumcircle in point MM. Find the locus of circumcenters of triangles AMKAMK.
geometryLocusLocus problemsCircumcentercircumcircle
given triangle of touchpoints of incircle has same orthocenter with ABC

Source: Sharygin 2010 Final 10.6

11/25/2018
The incircle of triangle ABCABC touches its sides in points A′,B′,C′A', B',C' . It is known that the orthocenters of triangles ABCABC and A′B′C′A' B'C' coincide. Is triangle ABCABC regular?
geometryincircleorthocenterright triangle