MathDB

2

Part of 2017 Ukrainian Geometry Olympiad

Problems(2)

equal angles related to a trapezoid

Source: Ukrainian Geometry Olympiad 2017, IX p2

12/11/2018
Point MM is the midpoint of the base BCBC of trapezoid ABCDABCD. On base ADAD, point PP is selected. Line PMPM intersects line DCDC at point QQ, and the perpendicular from PP on the bases intersects line BQBQ at point KK. Prove that ∠QBC=∠KDA\angle QBC = \angle KDA.
geometrytrapezoidequal anglesAngle Chasing
angle chasing, double angle given, perpendicularity wanted

Source: Ukrainian Geometry Olympiad 2017 XI p2

12/12/2018
On the side ACAC of a triangle ABCABC, let a KK be a point such that AK=2KCAK = 2KC and ∠ABK=2∠KBC\angle ABK = 2 \angle KBC. Let FF be the midpoint of ACAC, LL be the projection of AA on BKBK. Prove that FL⊥BCFL \bot BC.
geometryAngle Chasingperpendicularity