Mein gott what have you done to my diagram
by agbdmrbirdyface, Nov 6, 2016, 12:49 AM
I inverted it.
I don't know if it's great to write solutions with motivation involved, but that's what I do:
IMO 2015/3 wrote:
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.
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.I don't know if it's great to write solutions with motivation involved, but that's what I do:
![[asy]
size(250);
pair A, B, C, F, H, K, Q, L, M, O, N, T;
A = origin;
B = (17/14) * dir(310);
C = dir(250);
F = foot(A, B, C);
H = orthocenter(A, B, C);
path circumcircle = circumcircle(A, B, C);
O = circumcenter(A, B, C);
M = midpoint(B--C);
T = midpoint(H--A);
path ninepoint = circumcircle(F, M, T);
path circ1 = circle(T, distance(A, T));
pair[] intersection1 = intersectionpoints(circumcircle, circ1);
Q = intersection1[1];
path circ2 = circle(midpoint(H--Q), distance(H, Q)/2);
pair[] intersection2 = intersectionpoints(circumcircle, circ2);
K = intersection2[1];
N = circumcenter(Q, K, H);
path circQKH = circumcircle(Q, K, H);
path circFKM = circumcircle(K, F, M);
L = rotate(180, midpoint(H--O)) * N;
draw(A--B--C--cycle);
draw(circumcircle,red);
draw(ninepoint, red);
draw(A--F);
draw(A--Q--M);
draw(Q--K--L);
draw(circQKH,blue);
draw(circFKM,blue);
draw(M--N--T--L--cycle, orange);
draw(circumcircle(A, Q, L), dashed);
draw(M--L, dashed);
dot(H);
dot(F);
dot(M);
dot(O);
dot(T);
dot(Q);
dot(K);
dot(N);
dot(L);
label("$A$", A, dir(90));
label("$B$", B, SE);
label("$C$", C, SW);
label("$F$", F, S);
label("$H$", H, dir(245)* 1.8);
label("$O$", O, NE);
label("$M$", M, SE);
label("$Q$", Q, dir(130)*0.05);
label("$K$", K, W);
label("$N$", N, dir(190));
label("$T$", T, NW);
label("$L$", L, dir(30));
[/asy]](http://latex.artofproblemsolving.com/a/8/7/a87739d383bcf40ac51b7c4ba2556488440b39ea.png)
and 
.
and the circumcircle of 
,
,
,
be the midpoint of 
,
,
.
is a diameter of the nine-point circle, so then 
must be a rectangle, with a little extra effort from angle chasing. 
then must lie on 
,
,
concur at 
;
is the perpendicular bisector of 
,
May 2017
April 2017