Where no reference to length is made the word altitude is often employed to denote the indefinite line forming the perpendicular.

Hence a triangle has three altitudes, one to each side.

88°. Theorem.--The three altitudes of a triangle are con

and

EAF is one line and EA=AF.

Similarly, DCE is one line and DC=CE,

Now, AC is || to FD, the altitude to AC is 1 to FD and passes through B the middle point of FD.

(72°)

Therefore the altitude to AC is the right bisector of FD, and similarly the altitudes to AB and BC are the right bisectors of DE and EF respectively.

But the right bisectors of the sides of the ADEF are concurrent (86°), therefore the altitudes of the ABC are

concurrent.

q.e.d. Def. The point of concurrence of the altitudes of a triangle is the orthocentre of the triangle.

Cor. 1. If a triangle is acute-angled (77°, 5), the circumcentre and orthocentre both lie within the triangle.

2. If a triangle is obtuse-angled, the circumcentre and orthocentre both lie without the triangle.

3. If a triangle is right-angled, the circumcentre is at the middle point of the side opposite the right angle, and the orthocentre is the right-angled vertex.

Def. The side of a right-angled triangle opposite the right angle is called the hypothenuse.

D

89°. The definition of 80° admits of three different figures, viz.:

1. The normal quadrangle (1) in which each of the internal angles is less than a straight angle. When not

otherwise qualified the term quadrangle will mean this figure.

2. The quadrangle (2) in which one of the internal angles, as at D, is greater than a straight angle. Such an angle in a closed figure is called a re-entrant angle. We will call this an inverted quadrangle.

3. The quadrangle (3) in which two of the sides cross one another. This will be called a crossed quadrangle.

In each figure AC and BD are the diagonals, so that both diagonals are within in the normal quadrangle, one is within and one without in the inverted quadrangle, and both are without in the crossed quadrangle.

The general properties of the quadrangle are common to all three forms, these forms being only variations of a more general figure to be described hereafter.

90°. Theorem.-The sum of the internal angles of a quadrangle is four right angles, or a circumangle.

Proof. The angles of the two As ABD and CBD make up the internal angles of the quadrangle.

therefore the internal angles of the quadrangle are together equal to four right angles.

q.e.d.

Cor. This theorem applies to the inverted quadrangle as is readily seen.

91°. Theorem.-If two lines be respectively perpendicular to two other lines, the angle between the first two is equal or supplementary to the angle between the last two.

Then 4(BC. CD) is equal or supplementary to (AB. AD).

Ls at B and D are right angles :

BAD+2BCD=1,

B

E

Proof.-ABCD is a quadrangle, and the

D

(hyp.)

(90°)

(39°)

q.c.d.

or

But

BCD is supplementary to BAD.

LBCD is supplementary to LECD;

and the 4(BC, CD) is either the angle BCD or DCE. 4(BC. CD) is = or supplementary to ▲BAD.

EXERCISES.

1. ABC is a ▲, and A', B', C' are the vertices of equilateral As described outwards upon the sides BC, CA, and AB respectively. Then AA'-BB'=CC'. (Use 52°.) 2. Is Ex. I true when the equilateral As are described "inwardly" or upon the other sides of their bases? 3. Two lines which are parallel to the same line are parallel to one another.

4. L' and M' are two lines respectively parallel to L and M. The 4(L'. M')=≤(L. M).

5. On a given line only two points can be equidistant from a given point. How are they situated with respect to the perpendicular from the given point?

6. Any side of a ▲ is greater than the difference between the other two sides.

7. The sum of the segments from any point within a ▲ to the three vertices is less than the perimeter of the ▲. 8. ABC is a ▲ and P is a point within on the bisector of LA. Then the difference between PB and PC is less than that between AB and AC, unless the A is isosceles.

9. Is Ex. 8 true when the point P is without the ▲, but on the same bisector?

10. Examine Ex. 8 when P is on the external bisector of A, and modify the wording of the exercise accordingly. II. CE and CF are bisectors of the angle between AB and CD, and EF is parallel to AB. Show that EF is bisected by CD.

12. If the middle points of the sides of a ▲ be joined two and two, the ▲ is divided into four congruent As. 13. From any point in a side of an equilateral ▲ lines are drawn parallel to the other sides. The perimeter of the so formed is equal to twice a side of the ▲. 14. Examine Ex. 13 when the point is on a side produced.

15. The internal bisector of one angle of a ▲ and the external bisector of another angle meet at an angle which is equal to one-half the third angle of the ▲.

16. O is the orthocentre of the AABC. Express the angles AOB, BOC, and COA in terms of the angles A, B, and C.

17. P is the circumcentre of the ▲ABC. Express the angles APB, BPC, and CPA in terms of the angles A, B, and C.

18. The joins of the middle points of the opposite sides of any quadrangle bisect one another.

19. The median to the hypothenuse of a right-angled triangle is equal to one-half the hypothenuse.

20. If one diagonal of a ▲ be equal to a side of the figure, the other diagonal is greater than any side.

21. If any point other than the point of intersection of the diagonals be taken in a quadrangle, the sum of the

line-segments joining it with the vertices is greater than the sum of the diagonals.

22. If two right-angled ▲s have the hypothenuse and an acute angle in the one respectively equal to the like parts in the other, the As are congruent.

23. The bisectors of two adjacent angles of a are to one another.

24. ABC is a A. The angle between the external bisector of B and the side AC is (CA).

25. The external bisectors of B and C meet in D. Then 4BDC=(B+C).

26. A line L which coincides with the side AB of the AABC rotates about B until it coincides with BC, without at

any time crossing the triangle.

does it rotate?

Through what angle

27. The angle required in Ex. 26 is an external angle of the triangle. Show in this way that the sum of the three external angles of a triangle is a circumangle, and that the sum of the three internal angles is a straight angle. 28. What property of space is assumed in the proof of Ex. 27? 29. Prove 76° by assuming that AC rotates to AB by crossing the triangle in its rotation, and that AB rotates to CB, and finally CB rotates to CA in like manner.

SECTION V.

THE CIRCLE.

92°. Def. 1.—A Circle is the locus of a point which, moving

in the plane, keeps at a constant distance from a fixed point in the plane.

The compasses, whatever be their form, furnish us with two points, A and B, which, from the rigidity of the instrument, are supposed to preserve an unvarying distance

B

A

A

B

from one another. Then, if one of the points A is fixed, while the other B moves over the paper or other plane