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I have recently started doing some math for fun, and currently I'm having a dilemma.

triangles within triangles

The problem is as follows:

The two large triangles are equal in area. Which is larger; the area of pink triangles or the area of blue triangles? (in the image above)


That seems pretty straightforward. Blue, pink and large triangles are all similar (because they share 2 sides and the angle between them). There is a total of 9 (pink and white) triangles on the left image making the total area of pink triangles equal to $\frac{6}{9}$.

Now let's skip the blue triangles and try to get the area of pink triangles in a different way. If the pink triangles's both cathetuses length is $\frac{1}{3}$, then the area of a single pink triangle has to be $\frac{1}{18}$. Multiply that with the number of pink triangles and you get that the total area is $\frac{1}{3}$.

So I have a contradiction. One method of calculating area yields $\frac{2}{3}$ and another method for the same area gets me $\frac{1}{3}$.

Something is clearly incorrect, so my question is: why do the two methods give out two different results? What is wrong with my calculation?

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  • $\begingroup$ You forgot to calculate (and compare to) the area of the whole triangle, which is .... $\endgroup$ – Joffan Feb 27 '17 at 20:40
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The first calculation derives that the pink triangles cover $2/3$ of the total area of the triangle, that is $1/2$. So the area covered by the pink triangles is $2/3 \cdot 1/2 = 1/3$, which is what your second calculation gives you.

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