Equilateral triangles have interior angles of measure 60º. AL bisects angle MAY, so triangle ALY has angles 30º, 60º, and 90º. This means AY and LY occur in a ratio of √3 to 1. AY is half of AN, so AY = 9 and LY = 9/√3 = 3√3.
We can split up triangle MAN into 6 triangles with the same area as ALY, whose area is
1/2 * AY * LY = 1/2 * 9 * 3√3 = (27√3)/2
so that MAN has area
6 * (27√3)/2 = 81√3, or about 140.296.
Alternatively, we can observe that ML has the same length as AL, which by the Pythagorean theorem has length
AL = √(AY^2 + LY^2) = 6√3
Then MAN has area
1/2 * AN * (ML + LY) = 1/2 * 18 * (6√3 + 3√3) = 81√3
