Kicking the IMO P1 while it's down
by agbdmrbirdyface, Oct 9, 2016, 10:16 PM
Here's the really easy barycentric solution to this, and a better asymptote diagram:
http://www.artofproblemsolving.com/community/c282525h1314108
Solution:![[asy]
size(250);
pair A, B, C, J, K, L, M, F, G, S, T, I, O, P;
A = (0,0);
B = dir(250);
C = 1.5*dir(320);
draw(A--B--C--A, blue);
O = circumcenter(A, B, C);
I = incenter(A, B, C);
P = extension(O, midpoint(B--C), A, I);
J = -I + 2*P;
path Aexcircle = CP(J, foot(J, B, C));
draw(Aexcircle, red);
M = IP(Aexcircle, B--C);
K = extension(A, B, J, foot(J, A, B));
L = extension(A, C, J, foot(J, A, C));
draw(A--K, blue);
draw(A--L, blue);
F = extension(B, J, M, L);
draw(J--F--L, orange);
G = extension(C, J, M, K);
draw(K--G--J, orange);
S = extension(A, F, B, C);
draw(A--S, magenta);
draw(S--B, blue);
T = extension(A, G, B, C);
draw(A--T, magenta);
draw(T--C, blue);
dot(J);
dot(M);
dot(K);
dot(L);
dot(F);
dot(G);
dot(S);
dot(T);
label("$A$", A, N);
label("$B$", B, dir(215)*1.5);
label("$C$", C, dir(40)*1.5);
label("$J$", J, S);
label("$M$", M, N);
label("$K$", K, NW);
label("$L$", L, NE);
label("$F$", F, N);
label("$G$", G, N);
label("$S$", S, N);
label("$T$", T, N);
[/asy]](//latex.artofproblemsolving.com/1/e/3/1e3323cadbcfe9b15f9eb08e6744ee8b9dd04d6f.png)
Suppose we're not good at synth/proj geo (like me) and we're like what how do we approach this oh my lowd this looks rlly hard
Okay, so we can bary this. It looks like everything revolves around cevians and the point
, the A-excenter, which has really really nice coordinates in bary. So why not lol.
Let's list out the points needed to be calculated, and the order we should do them in. We know
, and can easily derive
but it's well-known. We can then easily get
, and then smash these together to get
, and then we can smash
together to get
. If everything goes well, we can easily see
is the midpoint here; otherwise, we can apply distance formula easily.
Let's begin!
We use
as our reference triangle, so
by areas. We conveniently know things about
that makes them easy to compute, because they're tangency points of the excircle to the lines
.
After these observations, we should have
. Wow that was easy.
On to computing
! We can see that line
is easy to compute because it's a "cevian"; the hard thing is to compute
. This line is given by the determinant
. Expanding gives us
. We note
has to be of the form
for some constant
, and we can plug this into
to get the coordinates we need.
. And after algebra we get
. Cool now we have
.
A similar thing happens to
and we get
. Now since
are both on line
we get
. Hmm it's not immediately obvious that these points are equidistant from
, but one look at the displacement vectors
gives us
and now it's clear that
is the midpoint of
, so we're done.
Tidbit
http://www.artofproblemsolving.com/community/c282525h1314108
Solution:
![[asy]
size(250);
pair A, B, C, J, K, L, M, F, G, S, T, I, O, P;
A = (0,0);
B = dir(250);
C = 1.5*dir(320);
draw(A--B--C--A, blue);
O = circumcenter(A, B, C);
I = incenter(A, B, C);
P = extension(O, midpoint(B--C), A, I);
J = -I + 2*P;
path Aexcircle = CP(J, foot(J, B, C));
draw(Aexcircle, red);
M = IP(Aexcircle, B--C);
K = extension(A, B, J, foot(J, A, B));
L = extension(A, C, J, foot(J, A, C));
draw(A--K, blue);
draw(A--L, blue);
F = extension(B, J, M, L);
draw(J--F--L, orange);
G = extension(C, J, M, K);
draw(K--G--J, orange);
S = extension(A, F, B, C);
draw(A--S, magenta);
draw(S--B, blue);
T = extension(A, G, B, C);
draw(A--T, magenta);
draw(T--C, blue);
dot(J);
dot(M);
dot(K);
dot(L);
dot(F);
dot(G);
dot(S);
dot(T);
label("$A$", A, N);
label("$B$", B, dir(215)*1.5);
label("$C$", C, dir(40)*1.5);
label("$J$", J, S);
label("$M$", M, N);
label("$K$", K, NW);
label("$L$", L, NE);
label("$F$", F, N);
label("$G$", G, N);
label("$S$", S, N);
label("$T$", T, N);
[/asy]](http://latex.artofproblemsolving.com/1/e/3/1e3323cadbcfe9b15f9eb08e6744ee8b9dd04d6f.png)
Suppose we're not good at synth/proj geo (like me) and we're like what how do we approach this oh my lowd this looks rlly hard
Okay, so we can bary this. It looks like everything revolves around cevians and the point

Let's list out the points needed to be calculated, and the order we should do them in. We know







Let's begin!
We use




After these observations, we should have

On to computing












A similar thing happens to










Tidbit
I should do 104 now because I'm actual trash at NT and I'm not doing anything to improve. Also, I'm trying to get onto the TJ Varsity Math Team and I should practice Duke and PuMAC problems oops.