Chapter 3

Prove Proposition \(\mathrm{I}-5\), that the base angles of an isosceles triangle are equal to one another.

Find a construction to bisect a given angle and prove that it is correct (Proposition I-9).

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.

Show that Proposition II-13 is equivalent to the law of cosines for an acute- angled triangle: In acute-angled triangles, the square on the side opposite the acute angle is less than the sum of the squares on the other two sides by twice the rectangle contained by one of the sides about the acute angle, namely, that on which the perpendicular falls, and the line segment between the angle and the perpendicular.

Prove Proposition III-3, that if a diameter of a circle bisects a chord, then it is perpendicular to the chord. And if a diameter is perpendicular to a chord, then it bisects the chord.

Provide the details of the proof of Proposition III-20: In a circle, the angle at the center is double the angle at the circumference, when the angles cut off the same arc.

Prove Proposition III-31, that the angle in a semicircle is a right angle.

Find a construction for circumscribing a circle about an arbitrary triangle.

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 the Euclidean algorithm to find the greatest common divisor of 963 and \(657 ;\) of 2689 and \(4001 .\)

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.

Prove Proposition \(\mathrm{V}-12\) both by using Eudoxus's definition and by modern methods: If any number of magnitudes are proportional, as one of the antecedents is to one of the consequents, so will all of the antecedents be to all of the consequents. (In algebraic notation, this says that if \(a_{1}: b_{1}=a_{2}: b_{2}=\cdots=a_{n}: b_{n}\), then \(\left(a_{1}+a_{2}+\cdots+a_{n}\right)\) \(\left.\left(b_{1}+b_{2}+\cdots+b_{n}\right)=a_{1}: b_{1}-\right)\)

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

Solve geometrically the equation \(\frac{9}{x}=\frac{x}{5}\) by beginning with a semicircle of diameter \(9+5=14\)

Prove Proposition VI-14, that in equal and equiangular parallelograms, the sides about the equal angles are reciprocally proportional and conversely.

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

Give a modern proof of the result that there are infinitely many prime numbers. Compare your proof to Euclid's and comment on the differences.

. Discuss the advantages and disadvantages of a geometric approach relative to a purely algebraic approach in the teaching of the quadratic equation in school.

Discuss whether Euclid's Elements fits Plato's dictums that the study of geometry is for "drawing the soul toward truth" and that it is to gain knowledge "of what eternally exists."