MathDB

4

Part of 2015 Iran Geometry Olympiad

Problems(2)

4 triangles with equal areas, prove MNPQ parallelogram

Source: IGO 2015 Elementary 4

7/22/2018
In rectangle ABCDABCD, the points M,N,P,QM,N,P, Q lie on ABAB, BCBC, CDCD, DADA respectively such that the area of triangles AQMAQM, BMNBMN, CNPCNP, DPQDPQ are equal. Prove that the quadrilateral MNPQMNPQ is parallelogram.
by Mahdi Etesami Fard
geometryareasparallelogramrectangle
Prove isoceles triangle given 12 points on 2 circles

Source: Iranian Geometry Olympiad (Advanced) 2015 Q4

11/5/2017
In triangle ABCABC, we draw the circle with center AA and radius ABAB. This circle intersects ACAC at two points. Also we draw the circle with center AA and radius ACAC and this circle intersects ABAB at two points. Denote these four points by A1,A2,A3,A4A_1, A_2, A_3, A_4. Find the points B1,B2,B3,B4B_1, B_2, B_3, B_4 and C1,C2,C3,C4C_1, C_2, C_3, C_4 similarly. Suppose that these 1212 points lie on two circles. Prove that the triangle ABCABC is isosceles.
geometry