The article presents several mathematical problems concerning Euclidean geometry

Yevdokimov, Oleksiy

English

University of Southern Queensland ePrints

The focus of this set of problems is Euclidean geometry. Since the ancienttimes, many magnificent results and theorems have been discovered andproved in this area of mathematics. Nevertheless, Euclidean geometry contin-ues to serve as a source of wonderful properties and interesting problems.The geometry problems here were all posed by Igor Sharygin, whose talent ingeometry problem posing is well known to mathematicians all over the world.Problem set 31. Let M be a point in the interior of quadrilateral ABCD such that AMBand CMD are isosceles triangles with the angle at vertex M equal to120°. Prove that there exists a point N such that triangles BNC and DNAare equilateral.2. Let ABCD be a cyclic quadrilateral such that CD = AD + BC. Prove thatangle bisectors at vertices A and B meet at the side CD.3. Let ABCD be a cyclic quadrilateral with perpendicular diagonals. Provethat mid-points of all sides and the perpendiculars dropped from thediagonal intersection point to the sides lie on a circle.4. Let AC be a diameter of a circle. Point E lies on AC. Draw chord BDthrough the point E in a way such that the area of quadrilateral ABCDwill have the greatest value.5. An angle bisector at vertex A of the parallelogram ABCD meets thestraight lines BC and CD at the points K and L respectively. Prove thatthe centre of a circle which contains points C, K and L lies on a circlewhich contains points B, C and D.63Australian Senior Mathematics Journal 23 (1)Ponder thisOleksiy Yevdokimov University of Southern Queensland<yevdokim@usq.edu.au>

Problem set 3

http://eprints.usq.edu.au/7861/1/Yevdokimov_ASMJ_v23n1_PV.pdf

Problem set 3

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University of Southern Queensland ePrints

University of Southern Queensland ePrints