Proposition 35. Problem.

272. To draw an escribed circle of a given triangle.

Given, the AABC.

Required, to describe a O touching AB, and CA, CB produced.

Cons. Bisect the s BAE, ABD by the lines AO, BO, which intersect at 0. (255)

From O draw OG, OH, OK I to CE, AB, CD.

With centre O and radius

OG, describe the GHK.

The

O.

GHK is the required

B

Proof. Since the pt. O is on the bisector of /BAE, it is equally distant from BA and AE.

(160)

... the Is OG, OH, OK are equal.

... a described with centre O and radius OG will touch CE, AB, CD at the pts. G, H, K.

... the GHK is an escribed of the AABC.

Q.E.F.

SCH. In the same manner the centres O', O" of the other two escribed Os may be found.

Therefore there are in general four circles tangent to three intersecting straight lines.

Proposition 36. Problem.

273. To circumscribe a circle about a given triangle.

Given, the A ABC.

Required, to circumscribe a about the Δ ΑΒΕ.

Cons. Draw DO, EOL to AC, AB at their mid. pts. intersecting at O (284).

With O as a centre, and a radius = OA, describe a O.

This

is the one required passing through the vertices A, B, C.

Proof. Since the pt. O is in the

bisectors OD, OE of

AC, AB, it is equally distant from the pts. A, B, C. Every pt. in the bisector of a line is equidistant from the extremities of the line (66).

... a described with centre O and radius OA must pass through the pts. A, B, C, and is ... circumscribed about the AABC.

Q.E.F.

274. SCH. This construction is the same as that of describing a circumference through any three given points not in the same straight line, or of finding the centre of a given circle, or of a given arc.

NOTE. Since the perpendicular bisector of a chord passes through the centre of the circle (202), therefore,

The perpendicular bisectors of the sides of a triangle are concurrent, the point of intersection being the centre of the circle circumscribed about the triangle.

The centre of the circle circumscribed about a triangle is sometimes called its circum-centre.

EXERCISE.

Prove (1) if the given triangle be acute-angled, the centre of the circumscribe circle falls within it; (2) if it be a right triangle, the centre falls on the hypotenuse; (3) if it be an obtuse-angled triangle, the centre falls without the triangle.

Proposition 37. Problem.

275. On a given straight line, to describe a segment which shall contain a given angle.

With O as a centre, and OA as a radius, describe the Oce AHBF.

The segment AHB is the segment required.

Proof. Join OB and BH. Since O is in the AB, it is equally distant from the pts. A, B.

bisector of

(66)

.. a described with centre O and radius OA must pass

through B.

Also, since AD is to AH at its extremity,

NOTE.-In the particular case when the given angle C is a right angle, the segment required will be the semicircle described on the given st. line AB.

let

276. To draw a common tangent to two given circles. Given, the Os AR, BS, and

AR > BS.

Join AC, and produce it to meet the Oce of the given

O in D.

Draw BE || to AD, and join DE.

DE is a common tangent to the two given Os.

Q. E. F.

AR,

Since two tangents can be drawn from B to the therefore two common tangents may always be drawn to the given Os. These are called the direct common tangents.

When the given Os are external to each other and do not intersect, two more common tangents may be drawn, called the transverse common tangents.

(2) Required, to draw the transverse pair of common tangents.

Cons. and proof. With centre A and radius the sum of the radii of the two Os, describe a O, and complete the construction, and proof, as in (1).

E

EXERCISES.

THEOREMS.

1. If a straight line cut two concentric circles, the parts of it intercepted between the two circumferences are equal.

2. If one circle touch another internally at P, prove that the straight line joining the extremities of two parallel diameters of the circles, towards the same parts, passes through P.

3. In Ex. 2, if a chord AB of the larger circle touches the smaller one at C, prove that PC bisects the angle APB.

4. If two circles touch externally at P, prove that the straight line joining the extremities of two parallel diameters towards opposite parts, passes through P.

5. Two circles with centres A and B touch each other externally, and both of them touch another circle with centre O internally show that the perimeter of the triangle AOB is equal to the diameter of the third circle.

6. In two concentric circles any chord of the outer circle which touches the inner, is bisected at the point of contact. 7. If three circles touch one another externally in P, Q, R, and the chords PQ, PR of two of the circles be produced to meet the third circle again in S, T, prove that ST is a diameter.

8. Points P, Q, R on a circle, whose centre is O, are joined; OM, ON are drawn perpendicular to PQ, PR respectively join MN, and show that if the angle OMN