A triangle divided by three line segments. What fraction is blue?

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## Solution

The blue fraction is 2/7.

geometry puzzles

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- Post author By Rik D. Tangerman
- Post date August 5, 2022
- 6 Comments on An irregular tiling

## 6 replies on “An irregular tiling”

Solution with ROUTH’s Theorem

If an interior triangle has vertices given by barycentric coordinates

(a,b,c), (d,e,f), (g,h,i), then the fraction of the whole triangle is the 3×3 determinant formed from those 9 values.

In this case, the barycentric coordinates are (4/7,2/7,/1/7) and its cyclic rotations, so the 3×3 matrix is a circulant matrix formed by rotating those three. The determinant of a circulant matrix whose first row is (a,b,c) is just a^3+b^3+c^3 = 3abc. Substituting (4/7,2/7,1/7) gives the fraction 1/7.

Similarly, one shows that each of the smaller triangles has fraction 1/21, for a total area of 2/7 for all 4 triangles combined.

There was a typo: the determinant of the circulant matrix is

a^3+b^3+c^3 – 3abc

Solution without ROUTH’s Theorem

One more way

Even easier: