MathDB

1

Part of 2015 Costa Rica - Final Round

Problems(2)

<BAC=? DE = BC, lines altitudes intersect circumcircle

Source: OLCOMA Costa Rica National Olympiad, Final Round, 2015 3.1

9/23/2021
Let △ABC\vartriangle ABC be such that ∠BAC\angle BAC is acute. The line perpendicular on side ABAB from CC and the line perpendicular on ACAC from BB intersect the circumscribed circle of △ABC\vartriangle ABC at DD and EE respectively. If DE=BCDE = BC , calculate ∠BAC\angle BAC.
anglesgeometrycircumcircle
cyclic wanted, ABCD with _|_ diagonals related, reflections of intersections

Source: OLCOMA Costa Rica National Olympiad, Final Round, 2015 C2 p1

9/30/2021
Let ABCDABCD be a quadrilateral whose diagonals are perpendicular, and let SS be the intersection of those diagonals. Let K,L,MK, L, M and NN be the reflections of SS on the sides ABAB, BCBC, CDCD and DADA respectively. BNBN cuts the circumcircle of △SKN\vartriangle SKN at EE and BMBM cuts the circumcircle of △SLM\vartriangle SLM at FF. Prove that the quadrilateral EFLKEFLK is cyclic.
geometryreflectionCyclicConcyclic