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It looks like a rotation of this triangle is really helpful to an easy solution?  Would be 3, if it is indeed an isoceles triangle tipped over.  

 

It is an arbitrary triangle with arbitrary lines in it. No other symmetries assumed.

The three given areas uniquely determine the fourth area.

Like in the previous missing area puzzle (https://www.scienceforums.net/topic/131527-find-the-missing-area/?do=findComment&comment=1239116), it is helpful to switch to triangles:

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The area in question is x+y.

 

 

 

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7.8?

I made a right triangle that fits the specification, but it got pretty complicated. I haven't figured out how to use the clue!

 

@md65536  Hint:

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(4+2)/2 = (3+x+y)/x

 

Edited May 12, 20232 yr by Genady

I can't imagine ever noticing that, but now I see why it works for other triangles with the same areas. If you skew the triangle eg. to make a right-angle triangle, you don't change the areas. If you uniformly scale the triangle vertically by r (preserving the ratios between the areas), and then scale horizontally by 1/r, each region's area is scaled by the same factor.

16 minutes ago, md65536 said:

I can't imagine ever noticing that, but now I see why it works for other triangles with the same areas. If you skew the triangle eg. to make a right-angle triangle, you don't change the areas. If you uniformly scale the triangle vertically by r (preserving the ratios between the areas), and then scale horizontally by 1/r, each region's area is scaled by the same factor.

More rigorously,

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the big triangle with the area 4+2 and the small triangle with the area 2 have the same base. Thus, the ratio of their areas equals the ratio of their heights. The same holds for the ratio of the triangles 3+x+y and x.

Now, these triangles have common heights, i.e., the triangles 4+2 and 3+x+y share the height, and the triangles 2 and x do too. Hense the above ratios of their areas are equal.