Prove that the total area of the regions 1 and 2 below equals the area of the triangle. Regions 3 and 4 are parts of the semi-circle in which the triangle is inscribed, and and are diameters for the other two semi-circles.

Source: Julia Robinson Mathematics Festival

**PROOF**

Area 1 = area of semi-circle area 3

area 3

area 3 (1)

Area 2 = area of semi-circle area 4

area 4

area 4 (2)

Adding Eq. (1) and Eq. (2) yields

Area 1 + Area 2 = area 3 area 4

area 3 area 4

Triangle is a right triangle because inscribed angle intercepts an arc of .

By the Pythagorean theorem, . Hence, it follows that

Area 1 + Area 2 = area 3 area 4

The right hand-side of the above equation is precisely the area of triangle ABC.

DONE

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