# Beautiful Geometry 1

## A Rectangle Inside a Square

The blue rectangle covers half of the square’s area. What’s the angle?

**SOLUTION**

Let the square be ABCD and the rectangle be AEFG. Let EF and FG intersect BD in points J and K. We denote by x, y, z, T the lengths of DK, KJ, JB, BC respectively.

From S(ABD)=S(AEFG), we get that S(KDG)+S(BJE)=S(KJF), and since △KDG, △BJE, △KJF are 90°-45°-45°, we find that:

x^2+y^2=z^2.

Using the Pythagorean Theorem, we choose a point L inside △BCD, such that KL=x and KL=y, and △KJL is right-angle.

We have

∠DKL+∠LJB=360°-LKJ-KJL=270°,

so using that △JBL and △DKL are isosceles, we get:

∠BLD=90°+∠KLD+∠BLJ=270°-∠DKL-∠LJB=135°.

Thus, the point L lies on the circle with center A and radius AB, and AD=AL=AB. Then, we see that AK and AJ are bisectors of ∠DAL and ∠LAB respectively. We conclude:

∠KAJ=∠KAL+∠LAJ=\frac{∠DAL+∠LAB}{2}=45°.

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