Another two of those Catriona Shearer puzzles.
5.
Shear Beauty
The area of the bottom left square is 5. What’s the area of the blue triangle?
6.
All Men are Created Equilateral
Spoilers!
The first makes use of the fact the area of a triangle depends on the length of one side (the “base”) and the perpendicular distance of the opposite vertex to the line containing the base. Since it’s the perpendicular distance, that means you can move the opposite vertex anywhere on a line parallel to the base and get a triangle of equal area.
In this picture the diagonal (lower left to upper right) of the big square is parallel to the short side of the triangle, so the top right vertex of the triangle can be moved to anywhere on the square’s diagonal to get a triangle with the same area. In particular it can be moved to the lower left corner to get a triangle whose dimensions don’t depend on the size of the large square.
And the blue triangle’s area is the area of the 6-square rectangle formed by the small squares, minus half the yellow square (1/2 square), minus half the orange square (2 squares), minus half the 3-square rectangle along the bottom (3/2 square): 6-1/2-2-3/2 = 2 squares, and since one square has area 5, the triangle has area 10.
In the second, cut the purple triangle in half:
and you can see the white triangles are congruent to half the purple triangle. Draw some more lines:
A white triangle is half the red triangle plus half the yellow triangle. So the red triangle is two half red triangles which is two half purple triangles minus two half yellow triangles, or red triangle area = 20-5 = 15.
MathematRec 
for the equilateral one, are you assuming a right angle between the left edge of the red triangle and the base of the white triangle?
I might have made that assumption at the time — not sure, it was a couple years ago! But you don’t have to assume it. Since the area of the purple equilateral triangle (she doesn’t explicitly say they’re all equilateral but the puzzle title implies it) is four times the area of the yellow one its sides are twice as long. By symmetry all the white triangles are congruent, so the longest side of each is twice the length of the shortest side. But the angle between those two sides is 60° and the short leg of a 60° right triangle is half the hypotenuse, so the white triangles are indeed right.
At this point you don’t have to draw any lines though. Just note the area of the equilateral triangle on the hypotenuse is the sum of the areas of the equilateral triangles on the two legs (a consequence of Pythagoras) so red area is 20 – 5 = 15.
cool, thanks!