MathDB

2

Part of 2016 China National Olympiad

Problems(1)

Concylic implies concyclic

Source: China Mathematical Olympiad 2016 Q2

12/16/2015
In △AEF\triangle AEF, let BB and DD be on segments AEAE and AFAF respectively, and let EDED and FBFB intersect at CC. Define K,L,M,NK,L,M,N on segments AB,BC,CD,DAAB,BC,CD,DA such that AKKB=ADBC\frac{AK}{KB}=\frac{AD}{BC} and its cyclic equivalents. Let the incircle of △AEF\triangle AEF touch AE,AFAE,AF at S,TS,T respectively; let the incircle of △CEF\triangle CEF touch CE,CFCE,CF at U,VU,V respectively. Prove that K,L,M,NK,L,M,N concyclic implies S,T,U,VS,T,U,V concyclic.
geometrycircumscribed quadrilateral