the other of the circles used in the construction, and one common tangent can be drawn.

If one of the circles is wholly within the other, AB is less than the difference of their radii, and B lies within each of the circles used in the construction, and no common tangent can be drawn.

Ex. 100. circles are equal.

Give the construction of this Problem when the

EXERCISES.

ΙΟΙ. Draw a straight line whose extremities and points of trisection lie on the circumferences of two equal circles whích touch each other.

102. Two given circles touch each other externally at P and are touched by the straight line AB at A and B respectively; shew that the circle on AB as diameter passes through P and touches the line joining the centres of the given circles.

103. If three equal circles touch each other, their centres are the corners of an equilateral triangle. So also are the points of contact.

104. If two circles touch externally at E, and AB, CD be any two parallel diameters of the circles, shew that the straight lines AD, BC will pass through E.

105. If the radius of one circle is the diameter of another, any straight line drawn from the point of contact to the outer circumference is bisected by the interior one.

106. Draw a circle of given radius to touch a given circle

at a given point. When will there be only one solution?

107. Draw a circle with its centre at a given point to touch a

given circle. Shew that there are generally two solutions. When will there be only one?

108. Two circles touch externally.

Shew how to place

a line of given length so that it shall pass through the point of contact and have its extremities on the circumferences of the circles.

109. Two equal circles have a common chord AB. If a chord AC of one of them, equal to AB, when produced backwards passes through the centre of the other: shew that AB is equal to the radius of either circle.

IIO. Two circles touch externally. Shew that the square on the common tangent is equal to the rectangle contained by their diameters.

III. Given two circles and a point A on the circumference of one of them, draw a chord PA so that if it be produced to cut the other circle in Q, QA may be equal to PA.

SECTION VII.

INSCRIBED AND CIRCUMSCRIBED FIGURES.

PROB. 6. To describe a circle passing through three given points which are not in the same straight line.

The construction and proof are contained in Theorem 12.

PROB. 7. To describe the circles touching three given straight lines which intersect one another, but not in the same point.

Let AA', BB', CC' be three given straight lines which intersect but not in the same point :

it is required to describe a circle touching AA', BB', and CC.

B

E

F

A

Find the four points equidistant from AA', BB', and CC'.

Let O be any one of these points:

Inter. of Loci, ii.

draw OD perpendicular to AA', and describe a circle with centre

O and radius OD:

this shall be a circle touching the given lines.

Draw OE and OF perpendicular to BB', and CC';

then, because O is equidistant from AA', BB', and CC',

therefore OE and OF are each equal to OD;

therefore E and F lie on the circumference of the circle whose

centre is O and radius OD,

III. 1, Cor.

and the circle touches each of the lines AA', BB', CC', since the angles at D, E, F are right angles. III. 20.

Similarly with each of the other points as centre a circle may be drawn.

Therefore four circles touching AA', BB', and CC' have been drawn.

Q.E.F.

COR. If two parallel straight lines are intersected by a third straight line, two and only two circles can be drawn touching the three straight lines.

DEF. 15.

The circle that touches the three sides of a triangle is called the inscribed circle of the triangle.

DEF. 16. A circle that touches one side of a triangle and the other two sides produced is called an escribed circle of the triangle.

Ex. 112. If two parallel straight lines are intersected by a third straight line, shew that the circles touching the three straight lines are equal.

*Ex. 113. The line joining the centres of two escribed circles passes through an angular point of the triangle and is perpendicular to the line joining the centre of the inscribed to that of the third escribed circle.

Ex. 114.

The radii of the two equal escribed circles of an isosceles triangle are equal to the altitude of the triangle.

PROB. 8. In a given circle to inscribe a triangle equi angular to a given triangle.

Let ABC be the given circle, DEF the given triangle:

it is required to inscribe in ABC a triangle equiangular to DEF.

E

At any point A on the circumference of ABC draw the tangent GH. III. 20, Cor. 5. From A draw the chord AB making the angle HAB equal to the angle DEF,

I. Prob. 5.

and the chord AC making the angle GAC equal to the angle DFE.

Join BC:

then shall ABC be the triangle required.

III. 23.

The angle HAB is equal to the angle ACB in the alternate segment of the circle.

But the angle DEF is equal to the angle HAB;

therefore the angle ACB is equal to the angle DEF.

Ax. c.

I. 25.

In like manner the angle ABC is equal to the angle DFE; therefore the remaining angle BAC is equal to the remaining angle EDF; therefore the triangle ABC is equiangular to the triangle DEF, and it is inscribed in the circle ABC.

QE.F.