2012 Greece Team Selection Test
Given is an acute triangle $ABC$ $\left(AB<AC<BC\right)$,inscribed in circle $c(O,R)$.The perpendicular bisector of the angle bisector $AD$ $\left(D\in BC\right)$ intersects $c$ at $K,L$ ($K$ lies on the small arc $\overarc{AB}$).The circle $c_1(K,KA)$ intersects $c$ at $T$ and the circle $c_2(L,LA)$ intersects $c$ at $S$.Prove that $\angle{BAT}=\angle{CAS}$.



Diagram[asy]import graph; size(10cm); 

real labelscalefactor = 0.5; /* changes label-to-point distance */

pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ 

pen dotstyle = black; /* point style */ 

real xmin = -6.94236331697463, xmax = 15.849400903703716, ymin = -5.002235438802758, ymax = 7.893104843949444;  /* image dimensions */

pen aqaqaq = rgb(0.6274509803921569,0.6274509803921569,0.6274509803921569); pen uququq = rgb(0.25098039215686274,0.25098039215686274,0.25098039215686274); pen qqqqtt = rgb(0.,0.,0.2); 



draw((1.8318261909633622,3.572783369254345)--(0.,0.)--(6.,0.)--cycle, aqaqaq); 

draw(arc((1.8318261909633622,3.572783369254345),0.6426249310341638,-117.14497824050169,-101.88970202103212)--(1.8318261909633622,3.572783369254345)--cycle, qqqqtt); 

draw(arc((1.8318261909633622,3.572783369254345),0.6426249310341638,-55.85706977865775,-40.60179355918817)--(1.8318261909633622,3.572783369254345)--cycle, qqqqtt); 

 /* draw figures */

draw((1.8318261909633622,3.572783369254345)--(0.,0.), uququq); 

draw((0.,0.)--(6.,0.), uququq); 

draw((6.,0.)--(1.8318261909633622,3.572783369254345), uququq); 

draw(circle((3.,0.7178452373968209), 3.0846882800136055)); 

draw((2.5345020274407277,0.)--(1.8318261909633622,3.572783369254345)); 

draw(circle((-0.01850947366601585,1.3533783539547308), 2.889550258039566)); 

draw(circle((5.553011501106743,2.4491551634556963), 3.887127532933951)); 

draw((-0.01850947366601585,1.3533783539547308)--(5.553011501106743,2.4491551634556963), linetype("2 2")); 

draw((1.8318261909633622,3.572783369254345)--(0.7798408954511686,-1.423695174396108)); 

draw((1.8318261909633622,3.572783369254345)--(5.22015910454883,-1.4236951743961088)); 

 /* dots and labels */

dot((1.8318261909633622,3.572783369254345),linewidth(3.pt) + dotstyle); 

label("$A$", (1.5831274347452782,3.951671933606579), NE * labelscalefactor); 

dot((0.,0.),linewidth(3.pt) + dotstyle); 

label("$B$", (-0.6,0.05), NE * labelscalefactor); 

dot((6.,0.),linewidth(3.pt) + dotstyle); 

label("$C$", (6.188606107156787,0.07450151636712989), NE * labelscalefactor); 

dot((2.5345020274407277,0.),linewidth(3.pt) + dotstyle); 

label("$D$", (2.3,-0.7), NE * labelscalefactor); 

dot((-0.01850947366601585,1.3533783539547308),linewidth(3.pt) + dotstyle); 

label("$K$", (-0.3447473583572136,1.6382221818835927), NE * labelscalefactor); 

dot((5.553011501106743,2.4491551634556963),linewidth(3.pt) + dotstyle); 

label("$L$", (5.631664500260511,2.580738747400365), NE * labelscalefactor); 

dot((0.7798408954511686,-1.423695174396108),linewidth(3.pt) + dotstyle); 

label("$T$", (0.5977692071595602,-1.960477431907719), NE * labelscalefactor); 

dot((5.22015910454883,-1.4236951743961088),linewidth(3.pt) + dotstyle); 

label("$S$", (5.160406217502124,-1.8747941077698307), NE * labelscalefactor); 

clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); 

 /* end of picture */[/asy]