MathDB

3

Part of 2016 Iranian Geometry Olympiad

Problems(3)

Dissect triangle to N similar quadrilaterals

Source: Iranian Geometry Olympiad 2016 Medium 3

5/26/2017
Find all positive integers NN such that there exists a triangle which can be dissected into NN similar quadrilaterals.
Proposed by Nikolai Beluhov (Bulgaria) and Morteza Saghafian
geometry
2 parallelograms on consecutive sides of ABCD

Source: IGO Elementary 2016 3

7/22/2018
Suppose that ABCDABCD is a convex quadrilateral with no parallel sides. Make a parallelogram on each two consecutive sides. Show that among these 44 new points, there is only one point inside the quadrilateral ABCDABCD.
by Morteza Saghafian
geometryparallelogram
Iranian Geometry Olympiad (3)

Source: IGO 2016,Advanced level,P3

9/13/2016
In a convex qualrilateral ABCDABCD, let PP be the intersection point of ADAD and BCBC. Suppose that I1I_1 and I2I_2 are the incenters of triangles PABPAB and PDCPDC,respectively. Let OO be the circumcenter of PABPAB, and HH the orthocenter of PDCPDC. Show that the circumcircles of triangles AI1BAI_1B and DHCDHC are tangent together if and only if the circumcircles of triangles AOBAOB and DI2CDI_2C are tangent together. Proposed by Hooman Fattahimoghaddam
geometrygeometry proposed