Points M and P are chosen on the sides AB and BC of triangle ABC so that MP and AC are parallel, AC-MP = 2,

Points M and P are chosen on the sides AB and BC of triangle ABC so that MP and AC are parallel, AC-MP = 2, BP / PC = 3/2. Find the MP length.

Triangles ABC and MBP are similar, since all angles are equal (angle B is a common angle, angle M = A and angle P = C, which follows from the parallelism of lines MP and AC).
Then, we use the properties of similarity of triangles:
ВС / ВР = АС / МР = k.

BC side = ВР + РС and BP / PC = 3/2 by condition.
Then BP = 3/2 * PC, which means BC = BP + PC = 3/2 * PC + PC = 5/2 * PC.
Hence,
ВС / ВР = (5/2 * РС) / (3/2 * РС) = 5/3.
Then,
AC / MP = 5/3.
By condition AC – MP = 2.

We got a system of two equations with two unknowns.
AC = MP + 2;
AC / MP = (MP + 2) / MP = 5/3.
Hence,
3 * (MP + 2) = 5 * MP;
3 * MR + 6 = 5 * MR;
2 * MP = 6;
MP = 3.