5.2.4.32 Various Problems on the Plane

Condition: in the triangle \ (a b c \) points \ (m, n, k \) are located respectively on the sides \ (a b, b c, a c \) so that \ (a m: m b = 1: 2 \), \ (c n: n b = 1: 3, a k = k c \). Sifts \ (m n \) and \ (b k \) intersect at point \ (p \). Find about b trapezia \ (a b c d \) with the bases \ (a d \) and \ (b c \) such that \ (a d: b c = 5: 3 \), the diagonals intersect at point \ (m \). Express the vectors \ (\ overrightarrow {m a}, \ overrightarrow {m b}, \ overrightarrow {m c}, \ overritarrow {m d} \) through vectors \ ( \ vec {a} = \ overrightarrow {a b} \) and \ (\ vec {b} = \ overrightarrow {b c} \)