In the triangle ABC, the midpoints M and N of the sides BC and AC, respectively, are marked.

In the triangle ABC, the midpoints M and N of the sides BC and AC, respectively, are marked. The area of the triangle CNM is 57. Find the area of the quadrilateral ABM N.

Consider a triangle ABC.

M and N are midpoints of sides BC and AC. Means

BM = CM and AN = CN.

Then the area of the triangle CNM S1 = 0.5 * CN * CM * sin (C) and

area of triangle ABC S2 = 0.5 * AC * BC * sin (C).

Then

S2 = 0.5 * AC * BC * sin (C) = 0.5 * 2 * CN * 2 CM * sin (C) =

= 4 * 0.5 * CN * CM * sin (C) = 4 * S1.

Then the area of the quadrangle ABMN = S2 – S1 = 3 * S1 =

= 3 * 57 = 171.