219. CONSTRUCTION. To circumscribe a circle about a given triangle.

Draw the perpendicular bisectors of two of the sides BC and AC.

With their intersection as a centre, and the distance to any vertex as a radius, describe a circumference.

This gives the circle required.

Proof.--O is equally distant from B and C.)

O is equally distant from A and C.

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[The perpendicular bisector is the locus of points equally distant from the extremities of a straight line.]

Therefore is equally distant from all vertices, and the circle described as above is the required circle.

Q. E. D.

220. Remark.-The foregoing construction also enables us to draw a circumference through three points noi in the same straight line or to find the centre of a given circumfer

ence or arc.

$166 221. CONSTRUCTION. To construct a tangent to a given circle from a given point without.

O

GIVEN

TO CONSTRUCT

the circle O and the point A without.

from A a tangent to the circle.

Upon AO as a diameter construct a circumference intersecting the given circumference at X and X'.

222. CONSTRUCTION. Upon a given straight line to con struct a segment which shall contain a given angle.

GIVEN

the straight line AB and the angle m.

TO CONSTRUCT—a segment upon AB which shall contain an angle equal to m.

At A construct m' equal to m, and having AB as one of its sides.

$ 80

Draw AO perpendicular to AC, and DO perpendicularly bisecting AB.

With O, the intersection of these two lines, as a centre, and OA or OB as a radius, construct a segment APB. This is the segment required.

Proof.-CA is tangent to the circle.

[Being perpendicular to a radius at its extremity.]

Therefore

m' is measured by arc AB

$173

$ 205

But m" (any angle inscribed in the segment) is also meas

PROBLEMS OF DEMONSTRATION

223. Defs. Two circles are tangent which touch at but one point. They may be tangent internally, so that one circle is within the other; or externally, so that each is without the other.

224. Exercise. The straight line joining the centres of two circles tangent externally passes through the point of tangency.

Χ

T

Hint.-Suppose 00' not through T, and prove 00' greater than and also less than the sum of the radii.

225. Exercise.-The straight line joining the centres of two circles internally tangent passes through the point of tangency.

O'

Y

Hint. If not, prove the distance between centres greater than and also

less than the difference of the radii.

226. Defs. If each of two circles is entirely without the other, four common tangents can be drawn. Two of these are called external, and two internal. An external tangent is one such that the two circles lie on the same side of it; an internal tangent is one such that the two circles lie on opposite sides of it.

Question.-In case the two circles are themselves tangent externally, how

many common tangents of each kind can be drawn? In case the two circles overlap? In case they are tangent internally? In case one is within the other?

227. Exercise.-The two common external tangents to two circles meet the line joining their centres in the same point. Also the two common internal tangents meet the line of centres in the same point.

228. Exercise.-The sum of two opposite sides of a quadrilateral circumscribed about a circle is equal to the sum of the other two sides (§ 176).

229. Exercise.-The sum of two opposite angles of a quadrilateral inscribed in a circle is equal to the sum of the other two angles, and is equal to two right angles.