2

Part of 2015 Sharygin Geometry Olympiad

Problems(3)

circumcircle of A,B, and orthocenter, meets AC,BC at interior points, <C ?

Source: Sharygin Geometry Olympiad 2015 Final 8.2

A circle passing through

A, B

and the orthocenter of triangle

ABC

AC, BC

at their inner points. Prove that

60^o < \angle C < 90^o

geometryorthocenterangle

find a point of convex ABCD so that it's projections on sides are vertices of #

Source: Sharygin Geometry Olympiad 2015 Final 9.2

A convex quadrilateral is given. Using a compass and a ruler construct a point such that its projections to the sidelines of this quadrilateral are the vertices of a parallelogram.
(A. Zaslavsky)

geometryparallelogramconstructionprojection

Covering by triangle

Source: Sharygin geometry olympiad 2015, grade 10, Final Round, Problem 2

Prove that an arbitrary triangle with area

1

can be covered by an isosceles triangle with area less than

\sqrt{2}