A rectangle with a diameter and two inscribed squares. What is its area in terms of the square areas A and B?

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## Solution

The rectangle area is B[1+√(B/(B-A))].

The rectangle side lengths are √X and √Y. The key is that all triangles are similar. It is then a matter of expressing their area in terms of the rectangle area T, as shown in the diagram.

From the lower half of the rectangle we get 1+T/2X+T/2Y=T/2B, whereas the upper half gives 1+T/2X+T/2Y+T/2(X+Y)=T/2A.

Subtracting these equations gives 1/(X+Y)=1/A-1/B, which leads to T^{2}(1/X+1/Y)=AB/(B-A).

Adding the two equations and using the above leads to T^{2}/B-2T-AB/(B-A)=0. This is a quadratic equation in T, which can be solved to find the desired answer.

## Poem

The letter box

Is our confidant and our friend

Especially in Xmas Time

All children are writing letters

Children’s love letters

Writing poems

Send drawings

Tell one’s story

Letters to get presents

To see the toys

To see Santas sleigh

The letter box is precious and usefull