Given is an acute triangle ABC (AB<AC<BC),inscribed in circle c(O,R).The perpendicular bisector of the angle bisector AD (D∈BC) intersects c at K,L (K lies on the small arc \overarc{AB}).The circle c1(K,KA) intersects c at T and the circle c2(L,LA) intersects c at S.Prove that ∠BAT=∠CAS.[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] geometryperpendicular bisectorangle bisector