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A0 = {x, y} /. Flatten[Solve[{y == n x, y == l (x + 2 t)}, {x, y}]];
B0 = {x, y} /. Flatten[Solve[{y == m x, y == l (x + 2 t)}, {x, y}]];
C0 = {x, y} /. Flatten[Solve[{y == n x, y == k (x - 5 t)}, {x, y}]];
D0 = {x, y} /. Flatten[Solve[{y == m x, y == k (x - 5 t)}, {x, y}]];
E0 = {k, l} /.
Solve[{((3 A0 + C0)/4)[[2]] == 0, B0[[2]] == C0[[2]]}, {k,
l}][[2]];
A1 = A0 /. {k -> E0[[1]], l -> E0[[2]]};
B1 = B0 /. {k -> E0[[1]], l -> E0[[2]]};
C1 = C0 /. {k -> E0[[1]], l -> E0[[2]]};
D1 = D0 /. {k -> E0[[1]], l -> E0[[2]]};
t0 = t /. Solve[(A1[[1]] D1[[2]] - A1[[2]] D1[[1]])/2 == 10, t][[2]];
FullSimplify[
Abs[B0[[1]] C0[[2]] - B0[[2]] C0[[1]]]/2 /. {k -> E0[[1]],
l -> E0[[2]], t -> t0}]
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