Three squares are placed as shown. Given that the green areas are equal, what is the side length of the orange square?

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## Solution

The answer is n=21.

Naming the angle between the small and the large square α, we can deduce from the equality of areas: cosα=(n-1)/(n+8).

Now we apply the cosine rule in the triangle: n^{2}=(n-1)^{2}+(n+8)^{2}-2(n-1)(n+8)cosα. Getting rid of the cosα, we get n^{2}-18n-63=0. This quadratic equation has only one positive root: 21.

## Poem

Geometrical poet Belladonna started the new year thus:

Les voeux de l’année 2021

Inclus dans trois carrés

Des cartes vertes et oranges

Portant des messages

Des voeux , des louanges ,

Des messages écrits

D’espoir, d’amour de poésie ,

Dire l ‘existentiel

Dire ce qui est essentiel

Bella