Denote ADE by the foot of perpendicular line from A to line DE. Given concyclic points A,B,C,D,E,F, show that the three points determined by the lines AFDADE , BDEBEF , CEFCFD, and the three points determined by the lines DCADAB , EABEBC , FBCFCA are concyclic.
Problem
Source: 2018 Korea Winter Program Practice Test 1 #3
Tags: Simson line, Concyclic, geometry
07.01.2018 16:01
We can prove that these 6 points are on a circle whose center is the midpoint of the orthocenters of ABC and DEF.
07.01.2018 16:34
CZRorz wrote: We can prove that these 6 points are on a circle whose center is the midpoint of the orthocenters of ABC and DEF. And then, complex numbers makes it a number 1 problem.
07.01.2018 17:38
It is enough to prove that BEF,AEF,EABEAC∩FABFAC,AEFAED∩BEFBFD are cyclic (because from similar facts we get that EABEAC∩FABFAC is Miquel point of EF,AFDADE,BDEBEF,CEFCFD and so EABEAC∩FABFAC lies on the circumircle of a triangle formed by lines AEFADE,BEFBDE,CEFCDE). In fact we will prove that EAB,FAB,BEF,AEF,EABEAC∩FABFAC,AEFAED∩BEFBFD are cyclic. It's well-known that EAB,FAB,BEF,AEF cyclic, so it's enough to prove that EABEAC∩FABFAC∈(EABFABBEFAEF). And it is some angle chasing : we know that EABEAC,FABFAC goes through EAB,FAB and angle between them can be computed. Done P.S. I think that this problem has been already posted on mathlinks.
07.01.2018 21:23
toto1234567890 wrote: And then, complex numbers makes it a number 1 problem. This is weird point here: According to marking scheme of this exam, you cannot receive over 2 points if you don't prove some lemmae for using complex numbers here. There are some compactification issues, I think, and also some policies to recommend synthetic proofs..
08.01.2018 04:02
IsoLyS wrote: This is weird point here: According to marking scheme of this exam, you cannot receive over 2 points if you don't prove some lemmae for using complex numbers here. There are some compactification issues, I think, and also some policies to recommend synthetic proofs.. Well... I agree. That's why I use complex thingies as the last thing I could do.