PROPOSITION VII. THEOREM.

393. If a figure is symmetrical with respect to two axes at right angles to each other, it is symmetrical with respect to their intersection as a centre.

Let the figure A-H be symmetrical with respect to the axes XX' and YY', intersecting each other at right angles. at O.

To prove that A-II is symmetrical with respect to 0 as

a centre.

Let P be any point in the perimeter of A-H.

Draw PQ and PR perpendicular to XX and YY'. Produce PQ and PR to meet the perimeter of A-H at P' and P", and draw QR, OP', and OP".

Then, since A-II is symmetrical with respect to XX',

But PQ OR, and hence OR is equal and parallel to P'Q.

=

Therefore, OP'QR is a parallelogram.

($ 109.)

Whence, QR is equal and parallel to OP'.

(§ 104.)

In like manner, we may prove OP"RQ a parallelogram;

and therefore QR is equal and parallel to OP".

Hence, since both OP' and OP" are equal and parallel to QR, P'OP" is a straight line which is bisected at 0.

That is, every straight line drawn through O is bisected at that point; whence, A-H is symmetrical with respect to O as a centre. ($ 390.)

ADDITIONAL EXERCISES.

BOOK I.

1. The bisectors of the exterior angles of a triangle form a triangle whose angles are respectively the half-sums of the angles of the given triangle taken two and two.

2. If CD is the perpendicular from C to the side AB of the triangle ABC, and CE is the bisector of the angle C, prove that

is one-half the difference of the angles A and B.

DCE

3. The lines joining the middle points of the adjacent sides of a quadrilateral form a parallelogram whose perimeter is equal to the sum of the diagonals of the quadrilateral.

4. The lines joining the middle points of the opposite sides of a quadrilateral bisect each other.

5. The lines joining the middle points of the opposite sides of a quadrilateral bisect the line joining the middle points of the diagonals.

6. The line joining the middle points of the diagonals of a trapezoid is parallel to the bases and equal to one-half their difference.

7. If D is any point in the side AC of the triangle ABC, and E, F, G, and I are the middle points of AD, CD, BC, and AB respectively, prove that EFGHI is a parallelogram.

8. If E and G are the middle points of the sides AB and CD of the quadrilateral ABCD, and F and II the middle points of the diagonals AC and BD, prove that ▲ EFH=\\FGH.

9. If D and E are the middle points of the sides BC and AC of the triangle ABC, and AD be produced to F and BE to G making DF AD and EG BE, prove that the line FG passes through C. 10. If D is the middle point of the side BC of the triangle ABC, prove AD<} (AB+ AC).

11. The sum of the medians of a triangle is less than the perimeter, and greater than the semi-perimeter of the triangle. (Ex. 113, p. 69.)

12. If the bisectors of the interior angle at C and the exterior angle at B of the triangle ABC meet at D, prove BDC=}ZA.

13. If AD and BD are the bisectors of the exterior angles at the extremities of the hypotenuse of the right triangle ABC, and DE and DF are drawn perpendicular, respectively, to CA and CB produced, prove that CEDF is a square.

14. AD and BE are drawn from two of the vertices of a triangle ABC to the opposite sides, making ≤ BAD=2 ABE ; if AD = BE, prove that the triangle is isosceles.

15. If perpendiculars AE, BF, CG, and DH, be drawn from the vertices of a parallelogram ABCD to any line in its plane, not intersecting its surface, prove that AE+ CG BF + DH.

16. If CD is the bisector of the angle C of the triangle ABC, and DF be drawn parallel to AC meeting BC at E and the bisector of the angle exterior to C at F, prove that DE = EF.

17. If E and F are the middle points of the sides AB and AC of the triangle ABC, and AD is the perpendicular from A to BC, prove that EDF = LEAF.

18. If the median drawn from any vertex of a triangle is greater than, equal to, or less than one-half the opposite side, the angle at that vertex is acute, right, or obtuse.

19. Prove that the number of diagonals of a polygon of n sides is n (n − 3).

2

20. The sum of the medians of a triangle is greater than threefourths the perimeter of the triangle.

21. If the lower base AD of a trapezoid ABCD is double the upper base BC, and the diagonals intersect at E, prove that CE AC and BE = | BD.

22. If O is the point of intersection of the bisectors of the angles of the equilateral triangle ABC, and OD and OE be drawn respectively perpendicular to BC and parallel to AC, meeting BC at D and E, prove that DE = BC.

23. If an equiangular triangle be constructed on each side of a triangle, the lines drawn from their outer vertices to the opposite vertices of the triangle are equal.

24. If two of the medians of a triangle are equal, the triangle is isosceles.

BOOK II.

25. AB and AC are the tangents to a circle from the point A, and D is any point in the smaller of the two arcs subtended by BC. If a tangent to the circle at D meets AB at E and AC at F, prove that the perimeter of the triangle AEF is constant.

26. The line joining the middle points of the arcs subtended by the sides AB and AC of an inscribed triangle ABC cuts AB at F and AC at G. Prove that AF= AG.

27. If ABCD is a circumscribed quadrilateral, prove that the angle between the lines joining the opposite points of contact is equal to (A + C).

28. If the sides AB and BC of an inscribed hexagon ABCDEF are parallel to the sides DE and EF respectively, prove that the side AF is parallel to CD.

29. If AB is the common chord of two intersecting circles, and AC and AD are the diameters drawn from A, prove that the line CD passes through B.

30. If AB is a common tangent to two circles which touch each other externally at C, prove that ACB is a right angle.

31. If AB and AC are the tangents to a circle from the point A, and D is any point on the circumference without the triangle ABC, prove that the sum of the angles ABD and ACD is constant.

32. If A, C, B, and D are four points in a straight line, B being between C and D, and EF is a common tangent to the circles described upon AB and CD as diameters, prove that ▲ BAE=<DCF.

33. ABCD is an inscribed quadrilateral, AD being a diameter of the circle. If O is the centre, and the sides AD and BC produced meet at E making CE = OA, prove that ZAOB 3 Z CED.

34. If ABCD is an inscribed quadrilateral, and its sides AD and BC are produced to meet at P, the tangent at P to the circle circumscribed about the triangle ABP is parallel to CD.

35. ABCD is a quadrilateral inscribed in a circle. If the sides AB and DC produced intersect at E, and the sides AD and BC produced at F, prove that the bisectors of the angles E and F are perpendicular to each other.

36. ABCD is a quadrilateral inscribed in a circle. Another circle is described upon AD as a chord, meeting AB and CD at E and F. Prove that the chords BC and EF are parallel.

37. If ABCDEFGH is an inscribed octagon, the sum of the angles A, C, E, and G is equal to six right angles.

38. If the number of sides of an inscribed polygon is even, the sum of the alternate angles is equal to as many right angles as the polygon has sides less two.

39. If the opposite angles of a quadrilateral are supplementary, the quadrilateral can be inscribed in a circle.

40. The perpendiculars from the vertices of a triangle to the opposite sides are the bisectors of the angles of the triangle formed by joining the feet of the perpendiculars. (Ex. 39.)

CONSTRUCTIONS.

41. Given a side, an adjacent angle, and the radius of the circumscribed circle of a triangle, to construct the triangle.

42. To describe a circle of given radius tangent to a given circle and passing through a given point.

43. Given an angle of a triangle, its bisector, and the length of the perpendicular from its vertex to the opposite side, to construct the triangle.

44. To draw between two given intersecting lines a straight line which shall be equal to one given straight line, and parallel to another.

45. Given an angle of a triangle, and the segments of the opposite side made by the perpendicular from its vertex, to construct the triangle.

46. To draw a parallel to the side BC of the triangle ABC meeting AB and AC in D and E, so that DE may be equal to EC.

47. To draw a parallel to the side BC of the triangle ABC meeting AB and AC in D and E, so that DE may be equal to the sum of BD and CE.

48. Given an angle of a triangle, the perpendicular from the vertex of another angle to the opposite side, and the radius of the circumscribed circle, to construct the triangle.

49. Given the base of a triangle, an adjacent angle, and the sum of the other two sides, to construct the triangle.

50. Given the base of a triangle, an adjacent acute angle, and the difference of the other two sides, to construct the triangle.

51. Through a given point without a given circle to draw a secant whose internal and external segments shall be equal. (Ex. 67, p. 103.)

52. Given the feet of the perpendiculars from the vertices of a triangle to the opposite sides, to construct the triangle. (Ex. 40.)

BOOK III.

53. State and prove the converse of Prop. XXVI., III.

54. In any triangle, the product of any two sides is equal to the product of the segments of the third side formed by the bisector of the exterior angle at the opposite vertex, minus the square of the bisector. (§ 288.)

55. If the sides of a triangle are AB

=

4, AC 5, and BC = 6,

find the length of the bisector of the exterior angle at the vertex A. (§ 250.)