91Ó°ÊÓ

Prove Proposition I-15, that if two straight lines cut one another, they make the vertical angles equal to one another.

Construct a triangle out of three given straight lines and prove that your construction is correct. Note that it is necessary that two of the straight lines taken together in any manner should be greater than the remaining one (Proposition I-22).

Draw a geometric diagram that proves the truth of Proposition II-8: If a straight line is cut at random, four times the rectangle contained by the whole and one of the segments together with the square on the remaining segment is equal to the square on the whole and the former segment taken together. Then translate this result into algebraic notation and verify it algebraically.

Find a construction for inscribing a regular hexagon in a circle.

Prove that the last nonzero remainder in the Euclidean algorithm applied to the numbers \(a, b\), is in fact the greatest common divisor of \(a\) and \(b\).

Use Theaetetus's definition of equal ratio to show that 33 : \(12=11: 4\) and that each can be represented by the sequence \((2,1,3)\)

Suppose that a line of length 1 is divided in extreme and mean ratio, that is, that the line is divided at \(x\) so that \(\frac{1}{x}=\) \(\frac{x}{x-1}\). Show by the method of the Euclidean algorithm that 1 and \(x\) are incommensurable. In fact, show that \(1: x\) can be expressed using Theaetetus's definition as \((1,1,1, \ldots)\).

Show that the side and diagonal of a square are incommensurable by using the method of anthyphairesis. Show that the ratio \(d: s\) can be expressed using Theaetetus's definition. as \((1,2,2,2, \ldots)\). Hint: Draw the diagonal of the square; then cut off on it the side and draw a square on the remaining segment.

. Construct geometrically the solution of \(8: 4=6: x\).

Prove Proposition VIII-14: If \(a^{2}\) measures \(b^{2}\), then \(a\) measures \(b\) and conversely.