MathDB

4

Part of 2020 Ukrainian Geometry Olympiad - April

Problems(2)

2<BCQ =<AQP in a square, <QPN = <NCB, NC=NP (Ukr. Geom. Olympiad '20 VIII p4)

Source:

6/8/2020
On the sides ABAB and ADAD of the square ABCDABCD, the points NN and PP are selected respectively such that NC=NPNC=NP. The point QQ is chosen on the segment ANAN so that ∠QPN=∠NCB\angle QPN = \angle NCB. Prove that 2∠BCQ=∠AQP2\angle BCQ = \angle AQP.
geometryequal segmentsequal anglesanglessquare
angle bisector wanted, <PAB = <PCB =1/4 (<A+ < C)

Source: Ukrainian Geometry Olympiad 2020, X p4

4/27/2020
Inside triangle ABCABC, the point PP is chosen such that ∠PAB=∠PCB=14(∠A+∠C)\angle PAB = \angle PCB =\frac14 (\angle A+ \angle C). Let BLBL be the bisector of △ABC\vartriangle ABC. Line PLPL intersects the circumcircle of △APC\vartriangle APC at point QQ. Prove that the line QBQB is the bisector of ∠AQC\angle AQC.
geometryangle bisectorequal angles