would have three equal lines drawn from the same point to the same straight line, which is impossible.

Therefore, it is false that A B intersects in a third point; therefore, it intersects in only two.

Q. E. D.

THEOREM XI.

A straight line perpendicular to a radius at its extremity, is tangent to the circumference at that point.

Let C be a circle, and A

AB a straight line perpendicular to the radius

CD at its extremity; then will A B have only this point in common with the circumference.

For, suppose it to touch

D

E

B

in another point, as E. Draw CE. Since, by supposition, E is on the circumference, CE CD; ... CDE CED (Ch. II, Th. VI), which is impossible (Ch. II, Th. XV).

=

=

Therefore, A B has no other point than D in common with the circumference, and is therefore tangent at D.

Q. E. D.

Cor. Conversely, a straight line tangent to the circumference at any point, is perpendicular to the radius drawn to that point.

For, any line, as CE, drawn from the centre to any other point of AB than the point of contact, is longer than CD, a radius; therefore CD is the shortest distance from C to AB; therefore CD is perpendicular to A B; and therefore AB is perpendicular to CD.

5

Scholium 1.-This proposition is evidently true whatever be the position of the parallel lines,-whether they are both secants, or one or both are tangents. Suppose E F to remain fixed, and let A B move toward D, but in such a manner as to continue parallel to E F. It is plain that in every position of A B, m n = op, and this will be true when n and o coincide with D, or when A B becomes tangent. If we suppose EF, in turn, to move downward till it reaches the point of tangency, it is equally plain that the two arcs continue equal.

Scholium 2.-The tangent may be regarded as a secant whose two points of intersection are coincident.

Note. The student should realize, at every step, that definitions, axioms, theorems, and corollaries, are the tools with which he is to work,-the materials, the instruments, upon which he must depend in the performance of a given task. Without them he is powerless. In proportion as they are held distinctly and securely in memory- are available, will be his satisfaction and success in the solution of problems and the demonstration of theorems.

EXERCISES.

1. In the same circle or in equal circles, chords equally distant from the centre are equal; and, of two chords unequally distant from the centre, that is the greater, whose distance from the centre is the less. (Ths. III and VIII.)

2. The least chord that can be drawn in a circle through a given point is the chord perpendicular to the line joining the given point and the centre.

3. There can be but one tangent at a given point of the circumference.

4. The normal at every point passes through the centre of the circle.

5. Any two tangents to a circle, drawn from a common point, are equal.

6. If from a point without the circumference, tangents be drawn to the extremities of a chord, the angle contained by the tangents is twice the angle contained by the chord and the diameter passing through either extremity of the chord.

7. A diameter is greater than any other chord.

8. The lines which bisect at right angles the sides of a triangle all meet in one point. (Th. IX.)

9. Find the locus of the centres of all circles that pass through two fixed points.

10. Find the locus of a point at a given distance from a given circle.

11. Find the locus of a point equidistant from two intersecting straight lines.

DEFINITIONS.

1. Two circles are concentric when they have the same

centre.

2. Two circles are coincident when they are concentric and equal.

3. Two circles are tangent externally when each is outside the other, with their circumferences in contact.

4. Two circles are tangent internally when one is within the other, with their circumferences in contact.

5. Two circles are external to each other when each is outside the other, with no point common.

6. One circle is wholly within another when it is enclosed by that other, with no point common to their circumferences.

7. Two circles intersect when they cut each other.

THEOREM XIII.

If two circles intersect each other, the straight line joining their centres bisects their common chord at right angles.

C

B

-C'

Let C and C' intersect in the points A and B, C C be the line joining their centres, and A B their common chord; then will C C' bisect A B at right angles.

For, the perpendicular drawn to A B at its middle point will pass through the centres, C and C', and therefore be coincident with the line joining these centres; hence CC' bisects A Bat right angles.

THEOREM XIV.

Q. E. D.

If two circles intersect, the distance between their centres is less than the sum, and greater than the difference of their radii.

B

C'

Let the circles, C and C', intersect

in the points A and B; then will

the distance between the centres be

less than the sum, and greater than the difference of the radii.

For, neither point of intersection is on C C', since C C' bisects the chord joining them, and therefore passes between them; hence, if either point of intersection be joined with the centres, the resulting figure is a triangle.

... CC' < CA + A C', and > CA - A C′ - Ch. II, Th. III.

THEOREM XV.

Q. E. D.

If two circles are externally tangent, the straight line joining their centres passes through the point of tangency.

Let C and C' be externally tangent at the point A; then will the straight line joining their centres pass through A.

For, through the point of contact draw the common tangent, D E, and to the same point draw the radii, and r'. Now, r and r are each perpendicular to D E at A (Th. IX), and hence form one and the same straight line (Ch.

r

D

E

II, Th. XIV), and the only straight line that can pass through the given centres (Ax. 7); hence, C, C', and A, are in the same straight line; hence, the line joining any two, as C and C', will pass through the third, as A.

Q. E. D.

Cor. I.-If two circles are externally tangent, the straight line joining their centres is perpendicular to their common tangent at the point of contact.

Cor. II.-If two circles are externally tangent, the distance between their centres is equal to the sum of their radii.

r