IMO 2015 Problem 3
by liberator, Jul 14, 2015, 2:48 PM
Problem: Let 
be an acute triangle with 
. Let 
be its cirumcircle, 
its orthocenter, and 
the foot of the altitude from 
. Let 
be the midpoint of 
. Let 
be the point on 
such that 
and let 
be the point on such that 
. Assume that the points , 
, 
, and are all different and lie on in this order.
Prove that the circumcircles of triangles
and 
are tangent to each other.
Proposed by Ukraine
be an acute triangle with 
. Let 
be its cirumcircle, 
its orthocenter, and 
the foot of the altitude from 
. Let 
be the midpoint of 
. Let 
be the point on 
such that 
and let 
be the point on such that 
. Assume that the points , 
, 
, and are all different and lie on in this order.Prove that the circumcircles of triangles
and 
are tangent to each other.Proposed by Ukraine
![[asy]
/* IMO 2015 Problem 3, free script by liberator, 14 July 2015 */
import olympiad;
import cse5;
unitsize(3.2cm);
pen ddd = rgb(0.4,0.6,0.8);
pointpen=black;
defaultpen(fontsize(10pt));
pair A = dir(75);
pair B = dir(205);
pair C = dir(-25);
pair O = origin;
pair H = orthocenter(A,B,C);
pair F = foot(A,B,C);
pair D = IP(L(H,F,0,2015),unitcircle);
pair M = midpoint(B--C);
pair Ap = B+C-H;
pair Q = IP(L(M,H,0,2015), unitcircle);
pair Qp = A+Ap-Q;
pair K = IP(L(Qp,H,-0.1,2015), unitcircle);
pair P = circumcenter(D,H,K);
D(unitcircle, heavygreen);
D(circumcircle(K,Q,H), red+linewidth(1));
D(circumcircle(F,K,M), heavymagenta);
D(circumcircle(D,H,K), grey+linetype("0 2")+linewidth(1));
D(A--B--C--cycle, ddd+linewidth(1));
D(A--D, ddd);
D(A--Qp, ddd);
D(Q--Ap, ddd+dashed);
D(K--Qp, ddd+dashed);
D(H--P--K, ddd);
D(rightanglemark(C,F,A,2), ddd);
D("A",A,dir(A)*2);
D("B",B,dir(B));
D("C",C,dir(C));
D("D",D,dir(-110));
D("F",F,dir(F));
D("H",H,dir(-65)*2);
D("K",K,dir(-15)*1.6);
D("M",M,dir(-70));
D("P",P,dir(P));
D("Q",Q,dir(Q));
D("A'",Ap,dir(Ap));
D("Q'",Qp,dir(Qp));
D("\Gamma",O,dir(160)*28);
[/asy]](http://latex.artofproblemsolving.com/a/a/6/aa6ae4ba390a833b56195faffc62aaf837297443.png)
be the antipodes of 
respectively w.r.t 
be the reflection of 
are collinear. Also observe that 
are collinear, since 
be the circumcenter of 
,
:
.
to 
at 
;
from rectangle 
,
,
,
is tangent to 
.
,
is tangent to 
:
,This post has been edited 2 times. Last edited by liberator, Jul 14, 2015, 8:58 PM
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