In the ABCD convex, general quadrilateral are given areas t₁, t₂, t₃. (t₃:t₂:t₁ = 15:12:6) Calculate t₄.

Note: M_AB is the midpoint between A and B.

Answer: 9

Here's how you solve it: consider the areas of pairs of triangles such as AMM_AB and BMM_AB that share an edge of the quadrilateral, and have a common altitude.  They are equal in area, because their bases are equal, and their heights are equal.

This way, the quadrilateral can be "colored" with four colors, so that

AMM_AB and BMM_AB are colored "a",
BMM_BC and CMM_BC are colored "b",
CMM_CD and DMM_CD are colored "c", and
DMM_AD and AMM_AD are colored "d"

Then we have these equations, which come from the areas of the colored triangles:

(1) a+d=15
(2) c+d=12
(3) b+c=6

By subtracting equation 2 from the sum of equation 1 and equation 3, we get

(4) a+b = 15+6-12 = 9