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P1 = {x1, y1} = {0, 0};
P2 = {x2, y2} = {x2, 0};
P3 = {x3, y3};
P = {xp, yp};
M = {{0, -1}, {1, 0}};

(* Umkreismittelpunkt *)
ls1 = Solve[(P1 + P2)/2 + s M.(P2 - P1) == (P1 + P3)/2 + t M.(P3 - P1), {s, t}];
Pm = (P1 + P2)/2 + s M.(P2 - P1) /. ls1[[1, 1]];

(* Schnittpunkte mit dem Umkreis *)
tt = -2 ((P - P1).(P1 - Pm))/((P - P1).(P - P1));
P1k = Simplify[P1 + tt (P - P1)];
tt = -2 ((P - P2).(P2 - Pm))/((P - P2).(P - P2));
P2k = Simplify[P2 + tt (P - P2)];
tt = -2 ((P - P3).(P3 - Pm))/((P - P3).(P - P3));
P3k = Simplify[P3 + tt (P - P3)];

(* Lote auf die Seiten *)
tt = ((P - P1).(P2 - P1))/((P2 - P1).(P2 - P1));
P3s = Simplify[P1 + tt (P2 - P1)];
tt = ((P - P1).(P3 - P1))/((P3 - P1).(P3 - P1));
P2s = Simplify[P1 + tt (P3 - P1)];
tt = ((P - P2).(P3 - P2))/((P3 - P2).(P3 - P2));
P1s = Simplify[P2 + tt (P3 - P2)];

(* Seiten und Winkel *)
bk = Simplify[P1k - P3k];
bs = Simplify[P1s - P3s];
ck = Simplify[P1k - P2k];
cs = Simplify[P1s - P2s];
cosak = Simplify[(bk.ck)/Sqrt[(bk.bk) (ck.ck)]];
cosas = Simplify[(bs.cs)/Sqrt[(bs.bs) (cs.cs)]];
Print[cosak^2];
Print[cosas^2];

(-x3^2 xp-xp y3^2+x2 (x3^2-x3 xp+y3 (y3-yp))+x3 (xp^2+yp^2))^2/((x3^2+y3^2) (x3^2-2 x3 xp+xp^2+(y3-yp)^2) (x2^2-2 x2 xp+xp^2+yp^2))

(-x3^2 xp-xp y3^2+x2 (x3^2-x3 xp+y3 (y3-yp))+x3 (xp^2+yp^2))^2/((x3^2+y3^2) (x3^2-2 x3 xp+xp^2+(y3-yp)^2) (x2^2-2 x2 xp+xp^2+yp^2))