MathDB

G3

Part of 2016 Costa Rica - Final Round

Problems(2)

collinear wanted, peprpendiculars around a rectangle

Source: OLCOMA Costa Rica National Olympiad, Final Round, 2016 Shortlist G3 day1

9/23/2021
Let the JHIZJHIZ be a rectangle and let AA and CC be points on the sides ZIZI and ZJZJ, respectively. The perpendicular from AA on CHCH intersects line HIHI at point XX and perpendicular from CC on AHAH intersects line HJHJ at point YY. Show that points X,YX, Y, and ZZ are collinear.
collineargeometryrectangle
6 points concyclic wanted, incircle and incenter related

Source: OLCOMA Costa Rica National Olympiad, Final Round, 2016 Shortlist G3 day2

9/25/2021
Let △ABC\vartriangle ABC be acute, with incircle Γ\Gamma and incenter I I. Γ\Gamma touches sides ABAB, BCBC and ACAC at ZZ, XX and YY, respectively. Let DD be the intersection of XZXZ with CICI and LL the intersection of BIBI with XYXY. Suppose DD and LL are outside of △ABC\vartriangle ABC. Prove that AA, DD, ZZ, II, YY, and L L lie on a circle.
geometryincenterConcyclic