MathDB

3

Part of 2014 European Mathematical Cup

Problems(2)

JEMC p3

Source: European Mathematical Cup 2014, Junior Division, Problem 3

12/22/2014
Let ABC be a triangle. The external and internal angle bisectors of ∠CAB intersect side BC at D and E, respectively. Let F be a point on the segment BC. The circumcircle of triangle ADF intersects AB and AC at I and J, respectively. Let N be the mid-point of IJ and H the foot of E on DN. Prove that E is the incenter of triangle AHF, or the center of the excircle.
Proposed by Steve Dinh
geometrycircumcircleincentergeometry unsolved
Parallelogram in cyclic quadrilateral

Source: European Mathematical Cup 2014, Senior Division, P3

12/14/2014
Let ABCDABCD be a cyclic quadrilateral in which internal angle bisectors ∠ABC\angle ABC and ∠ADC\angle ADC intersect on diagonal ACAC. Let MM be the midpoint of ACAC. Line parallel to BCBC which passes through DD cuts BMBM at EE and circle ABCDABCD in FF (F≠DF \neq D ). Prove that BCEFBCEF is parallelogram
Proposed by Steve Dinh
geometryparallelogramcircumcirclecyclic quadrilateralgeometry unsolved