## Verbal Description of the Greek Dissection of a Square

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**Verbal Description of the Greek Dissection of a Square**

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Suppose we are given two positive numbers a and b such that a ³ b, and a square whose side has length (a + b).

Let t1, t2, t3, and t4 be 4 points on the square such that:

t1 is on the upper edge, at a distance a from the upper left vertex;

t2 is on the right edge, at a distance a from the upper right vertex;

t3 is on the lower edge, at a distance a from the lower right vertex;

t4 is on the left edge, at a distance a from the lower left vertex.

Then the 4 line segments determined by the 4 pairs of points {t1, t2}, {t2, t3}, {t3, t4} and {t4, t1} determine a quadrilateral having 4 equal sides of length, say, c. By symmetry (or by complementary angles in the surrounding triangles) all of the angles of the quadrilateral are also equal. Therefore, the quadrilateral is a square, with area c2. The 4 line segments divide the square into 5 pieces: 4 right triangles, each having legs of lengths a and b, and hypotenuse of length c, and a square whose side has length c. Since the area of each right triangle is ((ab)/2), we have that the area of the original square is (4((ab/2)) + c^2).

This is called the Greek dissection of the square parameterized by (a,b).

Task: Draw the diagram corresponding to this discussion.

Note: The caret (^) is used instead of superscripting because the website word processor into which this article will be copied does not support superscripting.

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