Because of the symmetry of the set-up we are able to determine a few facts, irrespective of the particular shape of the triangle. We are able to assign areas A, B, C, D to different regions as shown. From the triangle SUY, A+B+C+D=1.

We can identify similar triangles PQR and PST, whose linear scaling is 1:2 and so whose area scaling is 1:4. So:

4B = A+2B+C

Similarly PUV has an area nine times that of PQR, so:

9B = 2A+2C+3B+2D

The first leads to 2B = A+C and the second leads to 3B = A+C+D, therefore B=D. Then, since B+D = A+C and all four add to 1, it follows that B and D are each equal to 1/4.

Next, look at similar triangles STX and QRW. The former is twice the linear scale, so four times the area of the latter, so:

C+D = 4C

3C = D

But we know that D=1/4, so C=1/12.

Since A+B+C+D=1, A = 1-1/4-1/12-1/4 = 5/12.

The area A is equal to five twelfths.