Denote $A_{DE}$ 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 $A_{FD}A_{DE}$ , $B_{DE}B_{EF}$ , $C_{EF}C_{FD}$, and the three points determined by the lines $D_{CA}D_{AB}$ , $E_{AB}E_{BC}$ , $F_{BC}F_{CA}$ 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 $B_{EF}, A_{EF}, E_{AB}E_{AC}\cap F_{AB}F_{AC}, A_{EF}A_{ED}\cap B_{EF}B_{FD}$ are cyclic (because from similar facts we get that $E_{AB}E_{AC}\cap F_{AB}F_{AC}$ is Miquel point of $EF, A_{FD}A_{DE}, B_{DE}B_{EF}, C_{EF}C_{FD}$ and so $E_{AB}E_{AC}\cap F_{AB}F_{AC}$ lies on the circumircle of a triangle formed by lines $A_{EF}A_{DE}, B_{EF}B_{DE}, C_{EF}C_{DE}$). In fact we will prove that $E_{AB}, F_{AB}, B_{EF}, A_{EF}, E_{AB}E_{AC}\cap F_{AB}F_{AC}, A_{EF}A_{ED}\cap B_{EF}B_{FD}$ are cyclic. It's well-known that $E_{AB}, F_{AB}, B_{EF}, A_{EF}$ cyclic, so it's enough to prove that $E_{AB}E_{AC}\cap F_{AB}F_{AC}\in (E_{AB}F_{AB}B_{EF}A_{EF})$. And it is some angle chasing : we know that $E_{AB}E_{AC}, F_{AB}F_{AC}$ goes through $E_{AB}, F_{AB}$ 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.