MathDB

4

Part of 1974 IMO Longlists

Problems(1)

Lines from excentres are concurrent [ILL 1974]

Source:

1/2/2011
Let Ka,Kb,KcK_a,K_b,K_c with centres Oa,Ob,OcO_a,O_b,O_c be the excircles of a triangle ABCABC, touching the interiors of the sides BC,CA,ABBC,CA,AB at points Ta,Tb,TcT_a,T_b,T_c respectively. Prove that the lines OaTa,ObTb,OcTcO_aT_a,O_bT_b,O_cT_c are concurrent in a point PP for which POa=POb=POc=2RPO_a=PO_b=PO_c=2R holds, where RR denotes the circumradius of ABCABC. Also prove that the circumcentre OO of ABCABC is the midpoint of the segment PIPI, where II is the incentre of ABCABC.
geometrycircumcircleincentergeometric transformationreflectiongeometry proposed