Perimeter of triangle $ABC$ is $1$. Circle $\omega$ touches side $BC$, continuation of side $AB$ at $P$ and continuation of side $AC$ in $Q$. Line through midpoints $AB$ and $AC$ intersects circumcircle of $APQ$ at $X$ and $Y$. Find length of $XY$.
Problem
Source: 44th International Tournament of Towns, Senior A-Level P2, Spring 2023
Tags: geometry
v_Enhance
05.04.2023 00:34
Let $O$ denote the center of $(APQ)$, and $I_a$ the $A$-excenter. Also, let $r_a$ denote the $A$-exradius. [asy][asy] pair A = dir(90); pair P = dir(230); pair Q = dir(310); pair I_a = dir(270); pair O = origin; pair X = dir(187); pair Y = dir(47); pair U = extension(X, Y, A, P); pair V = extension(X, Y, A, Q); pair B = 2*U-A; pair C = 2*V-A; pair T = foot(I_a, B, C); filldraw(unitcircle, invisible, blue); draw(P--A--Q, blue); filldraw(CP(I_a, T), invisible, red); pair N = extension(A, T, X, Y); draw(B--C, deepgreen); draw(X--Y, deepgreen); filldraw(CP(O, N), invisible, orange); dot("$A$", A, dir(A)); dot("$P$", P, dir(220)); dot("$Q$", Q, dir(320)); dot("$I_a$", I_a, dir(I_a)); dot("$O$", O, dir(45)); dot("$X$", X, dir(X)); dot("$Y$", Y, dir(Y)); dot("$B$", B, dir(160)); dot("$C$", C, dir(C)); dot("$T$", T, dir(110)); dot(N); /* -----------------------------------------------------------------+ | TSQX: by CJ Quines and Evan Chen | | https://github.com/vEnhance/dotfiles/blob/main/py-scripts/tsqx.py | +-------------------------------------------------------------------+ A = dir 90 P 220 = dir 230 Q 320 = dir 310 I_a = dir 270 O 45 = origin X = dir 187 Y = dir 47 U := extension X Y A P V := extension X Y A Q B 160 = 2*U-A C = 2*V-A T 110 = foot I_a B C unitcircle / 0.1 lightcyan / blue P--A--Q / blue CP I_a T / 0.1 lightred / red N .= extension A T X Y B--C / deepgreen X--Y / deepgreen CP O N / 0.1 yellow / orange */ [/asy][/asy] By homothety, $XY$ is tangent to the circle centered at $O$ with radius $r_a/2$. So the distance from $O$ to $AP$ and $AQ$ are also $r_a/2$ by the same homothety. So $XY=AP=AQ=1/2$.