5. To draw a tangent to a given circle making a given angle with a given line.

6. Any chord of a circle bisects the angle between the diameter through one extremity and the perpendicular from it on the tangent at the other.

7. Draw a circle through a given point to touch a given line at a given point. [Use 3 and 67.]

8. If a quadrangle is circumscribed about a circle, the sum of one pair of opposite sides is equal to the sum of the other pair.

9. If a convex quadrangle is such that the sum of one pair of opposite sides is equal to the sum of the other pair, then a circle may be inscribed in it.

81. Definitions.

TWO CIRCLES

Two circles are said to intersect at a point where they meet if they cross each other at this common point.

Two circles are said to touch at a point where they meet if they do not cross each other at this common point; and this point is called the point of contact.

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82. The line passing through the centers of two circles is called their central line.

POINTS COMMON TO TWO CIRCLES

Common point not on central line.

83. THEOREM 29. If two circles have one common point, not on their central line, then they have a second common point; and the circles intersect at each of these two points.

Let two circles whose centers are 0 and o' have the common point P, not on their central line.

First, to prove that they have a second common point. Draw PN perpendicular to 00', and prolong it to Q making NQ equal to PN. Draw OP, 0Q, O'P, O'Q.

By equality of triangles it follows that OP equals OQ, and O'P equals O'Q.

Therefore Q is a point on each of the circles (6).
Moreover, there is no third common point (22).

Next, to prove that the circles intersect at each of the points P, Q.

Let R, S be any two points on the circle whose center is o', situated at opposite sides of the point P. Draw OR, O'R, OS, O's.

In the triangles 00'R and 00'P, the sides oo' and O'R are respectively equal to the sides oo' and o'P, and the included angle 00'R is greater than 00'P.

Therefore the third side OR is greater than the third side OP (I. 91).

Therefore the point R is without the circle whose center is 0 (6).

In a similar way it is proved that os is less than OP. Therefore the point s is within the circle whose center is 0.

Now R and S are any two points on the circle whose center is O', situated at opposite sides of P.

Hence the two circles cross each other at P.

Similarly it can be proved that they cross at Q.

Common point on central line.

84. THEOREM 30. If two circles have one common point, situated on their central line, then they have no other common point; and the circles touch at this point.

Let 0, o' be the centers of the two circles, and P the common point on the central line 00'.

First, to prove that there is no other common point on the central line.

Suppose, if possible, that

is

R

another point on the line 00', common to the two circles. Then the segment PQ is a diameter of each circle. Hence the middle point of PQ is the center of each circle; which is impossible since the centers 0, o' do not coincide.

Therefore there is no second common point on the central

line.

Next, to prove that there is no second common point not on the central line.

Suppose, if possible, that R is another common point not on the central line. Then there is a third common point R', not on the central line (83).

Since there are three common points, the two circles coincide throughout (21).

This is contrary to the hypothesis; therefore there is no common point not on the central line.

Hence P is the only common point.

Again, to prove that the circles touch at P.

Let R be any other point on the circle whose center is o'. Draw OR and O'R.

Since O'R equals O'P, therefore the sum of O'R and 00' is equal to OP.

But OR is less than the sum of 00' and O'R (I. 87).

Therefore OR is less than OP.

Hence the point R is within the circle whose center is o and whose radius is OP.

Since R is any point (other than P) on the circle whose center is o', it follows that the circles do not cross at their common point P. Therefore they touch at this point

(81).

Point of contact.

85. THEOREM 31. If two circles touch, then their point of contact is on the central line; and they have no other common point.

Let there be two circles touching each other at a point.

To prove that the point of contact is on the central line; and that the circles have no other common point.

Suppose, if possible, that the point of contact is not on the central line.

Then, since the circles have a common point not on the central line, they intersect at this point (83).

This is contrary to the hypothesis; hence the supposition made is false. Therefore the point of contact is on the central line.

Further, since the two circles have a common point on their central line, it follows that they have no other common point (84).

86. Cor. I. If two circles touch each other externally, the line joining their centers is equal to the sum of their radii.

87. Cor. 2. If two circles touch, one being internal to the other, the line joining their centers is equal to the difference of their radii.

88. Cor. 3. If two circles do not meet, and each is wholly outside the other, the line joining their centers is greater than the sum of their radii.

89. Cor. 4. If two circles do not meet, and one is wholly inside the other, the line joining their centers is less than the difference of their radii.

90. Cor. 5. If two circles intersect, the line joining their centers is less than the sum of their radii, and greater than the difference of their radii.

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91. NOTE.

The relation of these five cases to each other is well shown by taking first the case in which each circle lies wholly outside the other, and then moving one center toward the other, as successively shown in the figures.

Ex. 1. In the five preceding corollaries, show that the hypotheses are exhaustive, and that the conclusions are mutually exclusive; then apply the rule of conversion (I. 104) to prove the converse of each corollary. E.g.,

If the line joining the centers of two circles is less than the difference of the radii, then one circle is wholly within the other.

Ex. 2. Find a point such that its joins to two given points may be equal respectively to two given lines. (Intersection of loci.)

Show when there are two solutions, when only one solution, and when none. Compare I. 132.

Ex. 3. If two circles touch, they have a common tangent at the point of contact.