Construction.

6. How does Euclid proceed, in order to get two points in the straight line A B equally distant from C? (See fig. page 117.)

7. Whereabouts in the straight line FG does it appear to you the foot of the perpendicular from C will come ?

8. Draw in pencil on the other side of FG all the construction required to get the point which will be the foot of the perpendicular drawn from C. Then efface all this construction, except the letter H.

9. What points do you now join to get the perpendicular required, and what do you say of the straight line CH?

10. What points have you to join for the demonstration?

Demonstration.

11. To prove that CH is perpendicular to A B what angles must be shown equal to each other?

12. What triangles have you to compare?

13. Which are the bases of those triangles? How do you know? 14. How do you know that the bases are equal?

15. What are the sides?

16. How do you know that FH is equal to HG?

17. Write the five steps of the demonstration required to show that the angle FHC is equal to G HC?

18. Having proved these angles equal, what can you say of CH? 19. Write out Proposition XII. as given in Euclid.

152

A SELECTION OF GEOMETRICAL EXERCISES REQUIRIN ONLY THE KNOWLEDGE OF THE FIRST TWELVE PRO POSITIONS OF THE FIRST BOOK OF EUCLID.

*** It is intended, and advised, that, throughout the following Exe cises, the constructions shall be done carefully, according to Euclid methods.

1. Given a straight line :-It is required to produce it so as to mak it altogether twice as long as the given straight line.

2. A straight line being given:-It is required, on that straight line as base, to describe an isosceles triangle, the sides of which shall be double the given base.

3. Given a straight line :—It is required to produce it so as to make it four times as long as the given straight line.

4. Given a straight line :-It is required to divide it into four equal parts.

5. Given a right angle:-It is required to divide it into four equal angles.

6. Given a straight line, and below it, to the right of it, a point A:It is required from A to draw a straight line equal to BC.

7. Do the same, when the given point is below the given straight line, to the left of it.

8. BAC is a given isosceles triangle of which BC is the base :-Bisect the vertical angle by a straight line which intersects the base in the point D.

9. The figure of exercise 8 being drawn:-Prove that BD is equal to DC; also that the straight line AD is at right angles to BC.

10. ABC is a given isosceles triangle of which BC is the base. Bisect AB in the point D, also bisect AC in the point E. Join DC and EB:-It is required to prove that DC and EB are equal.

11. ABC is a given equilateral triangle. Bisect AB in D, and AC in E. Let the perpendiculars to AB and AC drawn from D and E meet in the point F; join FA, FB, FC:-It is required to prove the straight lines FA, FB, and FC shall be all equal.

that

12. ABC is a triangle of which A is the vertex. It is given that ne perpendicular drawn from A, on the base, bisects the base :-It is equired to prove that the triangle ABC must be isosceles.

13. BC is a given straight line:-It is required on BC as base to ɔnstruct an isosceles triangle, of which the perpendicular height shall e equal to the base.

14. Draw the figure of Euclid's fifth proposition; and where the traight lines BG and FC intersect place the letter H:-It is required o prove that BH is equal to HC. Also that FH is equal to GH.

15. In the figure of Ex. 14 join AH:-Then prove that the angle 3AH is equal to the angle CAH.

16. Having proved that the angles BAH and CAH are equal Ex. 15):-Hence prove that AH bisects BC, and is at right angles to it.

17. ABC is an isosceles triangle standing on the base BC. Bisect the angles ABC and ACB by straight lines meeting in D:-It is required to prove that BDC is an isosceles triangle, of which BC is the base.

18. ABCD is a quadrilateral figure, in which it is given that the side AB is equal to the side AD. It is also given that the straight line AC bisects the angle BAD:-It is required to prove that BC is equal to DC; and that the angle BCD is bisected by the straight line AC.

19. ACB and AD B are two triangles on the same base AB and on the same side of it. It is given that AC is equal to BD, and that AD is equal to BC:-It is required to prove that the angle DBC is equal to the angle CAD.

20. If, in the figure to Ex. 19, AD and BC intersect in the point E:-Prove that the triangle A EB, standing on the base AB, is isosceles.

21. It is given that two isosceles triangles ABC and DBC stand on the same base, and on opposite sides of it; join AD:--It is required to prove, first, that the angle BAC is bisected by the straight line AD; and hence that AD bisects BC, and is at right angles to it.

22. ABCD is a rhombus (ie. a four-sided figure having all its sides equal); join AC:-It is required to prove that the angle BAD is bisected by AC. Prove also that the opposite angles BAD and BCD are equal.

23. ABC and DEF are two triangles, in which it is given (as in Prop. 8, quod vide), that the two sides BA, AC, are equal to the two sides ED, DF, each to each, and the base BC to the base EF. If B be placed on E, and BC be laid along EF, will C come on F? Why?— If now the triangle ABC be made to lie on the side of the common

I

base remote from D, and the vertices D and A be joined :-It is required to prove that the angle EAF (i.e. BAC) is equal to EDF.

[Obs. The foregoing is the proof given of the eighth prop., when Euclid's seventh proposition is omitted. To be done fully, three cases of the foregoing exercise should be considered. First, where both the angles at the base are acute; secondly, where one of them is obtuse; and lastly, where one of them is a right angle.]

24. ABCD is a given square; on one of its sides B C, as base stands an isosceles triangle EB C; join AE and DE. It is required to prove, first, that the angle ABE is equal to the angle DCE; secondly, that A E D is an isosceles triangle standing on the base AD.

25. The points O and P are the centres of two circles which intersect each other in the points A and B; join OP and AB:—It is required to prove that OP bisects A B, and is at right angles to it.

26. Let ABCD be a rectangle (that is, a figure having all its angles right angles, and its opposite sides equal); join BD:-It is required to prove that the two acute angles of a right-angled triangle are together equal to a right angle.

27. ABC is an equilateral triangle. In the sides A B, B C, C A the points D, E, F, are taken, so that A D, BE, CF are all equal. Join D E, EF, FD:-It is required to prove that D E F is an equilateral triangle.

28. From C any point in a straight line A B, CD is drawn at right angles to A B. With centre A and radius A B describe a circle intersecting CD in the point D. Join AD. From the straight line A D cut off A E equal to A C. Join EB-It is required to prove that AEB is a right angle.

29. CB is a straight line, of unlimited length, and A is a given point without it:—It is required to find in CB a point equally distant from A and B.

30. A B is a given straight line of unlimited length, and C, D two given points: It is required to find in the straight line AB a point equally distant from C and D. Show under what circumstances, in the 29th and 30th exercises, the solution is impossible, and under what circumstances the solutions are unlimited.

31. A and B are two points on opposite sides of the straight line CD-It is required to find a point K in CD such that if A K and BK be joined, the angles A K C and B K C shall be equal.

32. BC is the base of a triangle, D B C is one of the angles at the base, BD is equal to the sum of its two sides:-It is required to construct the triangle. [Obs. B D must be greater than B C; that it must be so, is proved in Prop. 20.]

155

HINTS, AND REFERENCES TO PROPOSITIONS BY WHICH THE EXERCISES CAN BE DONE.

Exercise

1. Postulates 2, 3, Definition 15.

2. Exercise 1 and Prop. 1.

3. As Ex. 1.

4. Prop. 10.

5. Prop. 9.

6,7. Varieties of the figure in Prop. 2.

8. Prop. 9.

9. Prop. 4.

10. Props. 10 and 4.

11. Props. 10, 11 and 4, and Ax. 1.

12. Prop. 4.

13. Props. 10, 11, 3 and 4.

14. Prop. 6, Ax. 3.

15. Prop. 8.

16. Prop. 4.

17. Props. 9 and 6.

18. Prop. 4.

19. Prop. 8.

20. Prop. 6.

21. Props. 8 and 4.

22. Part I., Prop. 8; Part II., Prop. 5.

23. Prop. 5 and Ax. 2.

24. Part I., Def. 30, Prop. 5 and Ax. 2 or 3; Part II. Prop. 4.

25. Same as Ex. 21.

26. Prop. 4.

27. Ax. 3, Prop. 4, and Ax. 1.

28. Props. 11 and 4.

29. Join AB, bisect it at right angles by a straight line, meeting CB in D. D will be the point required. Prop. 4.

30. Join CD. The rest as Ex. 29.

31. From B draw BE at right angles to CD. Produce BE to F, making EF equal to BE. Join AF and produce it to meet CD in K. Kis the required point. Prop. 4.

32. Join CD, bisect C D at right angles by a straight line meeting BD in A; join AC. A BC will be the triangle required. Prop. 4.