MathDB

5

Part of 2016 Iranian Geometry Olympiad

Problems(3)

AB = BC + AD in convex ABCD with many data given

Source: IGO Elementary 2016 5

7/22/2018
Let ABCDABCD be a convex quadrilateral with these properties: ∠ADC=135o\angle ADC = 135^o and ∠ADB−∠ABD=2∠DAB=4∠CBD\angle ADB - \angle ABD = 2\angle DAB = 4\angle CBD. If BC=2CDBC = \sqrt2 CD , prove that AB=BC+ADAB = BC + AD.
by Mahdi Etesami Fard
geometryanglesquadrilateral
Perpendicular bisector of secant tangent to circumcircle

Source: Iranian Geometry Olympiad 2016 Medium 5

5/26/2017
Let the circles ω\omega and ω′\omega' intersect in points AA and BB. The tangent to circle ω\omega at AA intersects ω′\omega' at CC and the tangent to circle ω′\omega' at AA intersects ω\omega at DD. Suppose that the internal bisector of ∠CAD\angle CAD intersects ω\omega and ω′\omega' at EE and FF, respectively, and the external bisector of ∠CAD\angle CAD intersects ω\omega and ω′\omega' at XX and YY, respectively. Prove that the perpendicular bisector of XYXY is tangent to the circumcircle of triangle BEFBEF.
Proposed by Mahdi Etesami Fard
geometryperpendicular bisectorcircumcircle
Iranian Geometry Olympiad (5)

Source: Advanced level,P5

9/13/2016
Do there exist six points X1,X2,Y1,Y2,Z1,Z2X_1,X_2,Y_1, Y_2,Z_1,Z_2 in the plane such that all of the triangles XiYjZkX_iY_jZ_k are similar for 1≤i,j,k≤21\leq i, j, k \leq 2? Proposed by Morteza Saghafian
geometrygeometry proposed