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

In the triangle ABC, the midpoints M and N of the sides BC and AC are marked, respectively. The area of the triangle CNM is 2. Find the area of the quadrilateral ABMN.

Consider triangle CNM.

By the condition of the problem CM = (1/2) * BC, CN = (1/2) * AC.

The area S1 of triangle CNM can be written:

S1 = 0.5 * CM * CN * sin (BCA).

The area S2 of triangle ABC can be written:

S2 = 0.5 * BC * AC * sin (BCA).

Then we have:

S2 = 0.5 * BC * AC * sin (BCA) = 0.5 * 2 * CM * 2 * CN * sin (BCA) = 4 * 0.5 * CM * CN * sin (BCA) = 4 * S1 …

The area S of the quadrilateral ABMN is:

S = S2 – S1 = 4 * S1 – S1 = 3 * S1 = 3 * 2 = 6.

Answer: The area of the quadrilateral ABMN is 6.