MathDB

G2

Part of 2016 Costa Rica - Final Round

Problems(2)

computational geo right triangle, AF _|_ EI, AI/IF=4/3

Source: OLCOMA Costa Rica National Olympiad, Final Round, 2016 Shortlist G2 day1

9/23/2021
Consider △ABC\vartriangle ABC right at B,FB, F a point such that B−F−CB - F - C and AFAF bisects ∠BAC\angle BAC, II a point such that A−I−FA - I - F and CI bisect ∠ACB\angle ACB, and EE a point such that A−E−CA- E - C and AF⊥EIAF \perp EI. If AB=4AB = 4 and AIIF=43\frac{AI}{IF}={4}{3} , determine AEAE.
Notation: A−B−CA-B-C means than points A,B,CA,B,C are collinear in that order i.e. B B lies between A A and CC.
geometryright triangle
AL + JB = LJ wanted, cyclic ABCD, angle bisectors

Source: OLCOMA Costa Rica National Olympiad, Final Round, 2016 Shortlist G2 day2

9/25/2021
Let ABCDABCD be a convex quadrilateral, such that A A, B B, CC, and DD lie on a circle, with ∠DAB<∠ABC\angle DAB < \angle ABC. Let II be the intersection of the bisector of ∠ABC\angle ABC with the bisector of ∠BAD\angle BAD. Let ℓ\ell be the parallel line to CDCD passing through point II. Suppose ℓ\ell cuts segments DADA and BCBC at L L and JJ, respectively. Prove that AL+JB=LJAL + JB = LJ.
geometrycyclic quadrilateral