Exercises.

1. If the angle ACB of a triangle be equal to twice the angle of an equilateral triangle, AB2 = BC2 + ČA2 + BC. CA.

2. ABCD is a quadrilateral whose opposite angles B and D are right, and AD, BC produced meet in E; prove AE. DE BE. CE.

3. ABC is a right-angled triangle, and BD is a perpendicular on the hypotenuse AC; Prove AB. DC BD. BC.

4. If a line AB be divided in C so that AC2 = 2CB2; prove that AB2+ BC2 2AB. AC.

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5. If AB be the diameter of a semicircle, find a point C in AB such that, joining C to a fixed point D in the circumference, and erecting a perpendicular CE meeting the circumference in E, CE2CD2 may be equal to a given square.

6. If the square of a line CD, drawn from the angle C of an equilateral triangle ABC to a point D in the side AB produced, be equal to 2AB2; prove that AD is cut in "extreme and mean ratio" at B.

PROP. XIII.—THEOREM.

In any triangle (ABC), the square on any side subtending an acute angle (C) is less than the sum of the squares on the sides containing that angle, by twice the rectangle (BC, CD) contained by either of them (BC) and the intercept (CD) between the acute angle and the foot of the perpendicular on it from the opposite angle.

Dem.-Because BC is divided into two segments

in D,

Therefore AB2 is less than BC2 + AC2 by 2BC. CD.

Or thus: Describe squares on the sides. Draw AE, BF, CG perpendicular to the sides; then, as in the demonstration of [I. XLVII.], the rectangle BG is equal

to BE; AG to AF, and CE to CF. Hence the sum of the squares on AC, CB exceeds the square on AB by twice CE—that is, by 2BC.CD.

Observation. By comparing the proofs of the pairs of Props. IV. and VII.; v. and VI.; Ix. and x.; XII. and XIII., it will be seen that they are virtually identical. In order to render this identity more apparent, we have made some slight alterations in the usual proofs. The pairs of Propositions thus grouped are considered in Modern Geometry not as distinct, but each pair is regarded as one Proposition.

Exercises,

1. If the angle C of the AACB be equal to an angle of an equilateral A, AB2 = AC2 + BC2 - AC. BC.

2. The sum of the squares on the diagonals of a quadrilateral, together with four times the square on the line joining their middle points, is equal to the sum of the squares on its sides. 3. Find a point C in a given line AB produced, so that AC2+ BC2 2AC. BC.

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PROP. XIV.-PROBLEM.

To construct a square equal to a given rectilineal figure (X).

Sol. Construct [I. XLV.] the rectangle AC equal to X. Then, if the adjacent sides AB, BC be equal, AC is a square, and the problem is solved; if not, produce AB to E, and make BE equal to BC; bisect AE in F;

with F as centre and FE as radius, describe the semicircle AGE; produce CB to meet it in G. The square described on BG will be equal to X.

Dem.-Join FG. Then because AE is divided equally in F and unequally in B, the rectangle AB. BE, together with FB2 is equal to FE2 [v.], that is, to FG2; but FG2 is equal to FB2 + BG2 [I. XLVII.]. Therefore the rectangle AB. BE + FB2 is equal to FB2 + BG2. Reject FB2, which is common, and we have the rectangle AB. BE BG2; but since BE is equal to BC, the rectangle AB. BE is equal to the figure AC. Therefore BG is equal to the figure AC, and therefore equal to the given rectilineal figure (X).

Cor. The square on the perpendicular from any point in a semicircle on the diameter is equal to the rectangle contained by the segments of the diameter.

Exercises.

1. Given the difference of the squares on two lines and their rectangle; find the lines.

2. Divide a given line, so that the rectangle contained by another given line and one segment may be equal to the square on the other segment.

Questions for Examination on Book II.

1. What is the subject-matter of Book II.? Ans. Theory of rectangles.

2. What is a rectangle? A gnomon?

3. What is a square inch? A square foot? A square perch? A square mile? Ans. The square described on a line whose length is an inch, a foot, a perch, &c.

4. What is the difference between linear and superficial measurement? Ans. Linear measurement has but one dimension; superficial has two.

5. When is a line said to be divided internally? When externally?

6. How is the area of a rectangle found?

7. How is a line divided so that the rectangle contained by its segments may be a maximum?

8. How is the area of a parallelogram found?

9. What is the altitude of a parallelogram whose base is 65 metres and area 1430 square metres ?

10. How is a line divided when the sum of the squares on its segments is a minimum ?

11. The area of a rectangle is 108.60 square metres and its perimeter is 48.20 linear metres; find its dimensions.

12. What Proposition in Book II. expresses the distributive law of multiplication?

13. On what proposition is the rule for extracting the square root founded?

14. Compare I. XLVII. and II. XII. and XIII.

15. If the sides of a triangle be expressed by x2 + 1, x2 − 1, and 2x linear units, respectively; prove that it is right-angled.

16. How would you construct a square whose area would be exactly an acre? Give a solution by I. XLVII.

17. What is meant by incommensurable lines? Give an example from Book II.

18. Prove that a side and the diagonal of a square are incommensurable.

19. The diagonals of a lozenge are 16 and 30 metres respectively; find the length of a side.

20. The diagonal of a rectangle is 4.25 perches, and its area is 7.50 square perches; what are its dimensions?

21. The three sides of a triangle are 8, 11, 15; prove that it has an obtuse angle.

22. The sides of a triangle are 13, 14, 15; find the lengths of its medians; also the lengths of its perpendiculars, and prove that all its angles are acute.

23. If the sides of a triangle be expressed by m2 + n2, m2 − n2, and 2mm linear units, respectively; prove that it is right-angled.

24. If on each side of a square containing 5.29 square perches we measure from the corners respectively a distance of 1.5 linear perches; find the area of the square formed by joining the points thus found.

Exercises on Book II.

1. The squares on the diagonals of a quadrilateral are together double the sum of the squares on the lines joining the middle points of opposite sides.

2. If the medians of a triangle intersect in O, AB2 + BC2 + CA2 = 3 (OA2 + OB2 + OC2).

3. Through a given point O draw three lines OA, OB, OC of given lengths, such that their extremities may be collinear, and that AB BC.

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4. If in any quadrilateral two opposite sides be bisected, the sum of the squares on the other two sides, together with the sum of the squares on the diagonals, is equal to the sum of the squares on the bisected sides, together with four times the square on the line joining the points of bisection.

5. If squares be described on the sides of any triangle, the sum of the squares on the lines joining the adjacent corners is equal to three times the sum of the squares on the sides of the triangle.