Doubly Trisected Triangle

Geometry Level 5

Triangle ABC has $$\angle A = 40^{\circ}$$, $$\angle B = 60^{\circ}$$, $$\angle C = 80^{\circ}$$. Points $$M,N$$ trisect the side $$BC$$ and points $$P,Q$$ trisect the side $$AC$$. The lines $$AM, AN, BP, BQ$$ intersect at the points $$S,T,U,V$$ as shown in the figure below, dividing the triangle into 9 regions. Determine the smallest possible value of $$[ABC] + [STUV]$$ such that both $$[ABC]$$ and $$[STUV]$$ are positive integers.

Details and assumptions

$$[PQRS]$$ refers to the area of figure $$PQRS$$.

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