EXERCISES.

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1. Draw a circle to pass through three given points.

2. If a circle touches a given line PQ at a point A, on what line must its centre lie?

If a circle passes through two given points A and B, on what line must its centre lie?

Hence draw a circle to touch a straight line PQ at the point A, and to pass through another given point B.

3. If a circle touches a given circle whose centre is C at the point A, on what line must its centre lie?

Draw a circle to touch the given circle (C) at the point A, and to pass through a given point B.

4. A point P is 4.5 cm. distant from a straight line AB. Draw two circles of radius 3.2 cm. to pass through P and to touch AB.

5. Given two circles of radius 30 cm. and 20 cm. respectively, their centres being 60 cm. apart; draw a circle of radius 3.5 cm. to touch each of the given circles externally.

How many solutions will there be? What is the radius of the smallest circle that touches each of the given circles externally?

6. If a circle touches two straight lines OA, OB, on what line must its centre lie?

Draw OA, OB, making an angle of 76°, and describe a circle of radius 1.2' to touch both lines.

7. Given a circle of radius 3.5 cm., with its centre 50 cm. from a given straight line AB; draw two circles of radius 2·5 cm. to touch the given circle and the line AB.

8. Devise a construction for drawing a circle to touch each of two parallel straight lines and a transversal.

Shew that two such circles can be drawn, and that they are equal.

9. Describe a circle to touch a given circle, and also to touch a given straight line at a given point. [See page 311.]

10. Describe a circle to touch a given straight line, and to touch a given circle at a given point.

11. Shew how to draw a circle to touch each of three given straight lines of which no two are parallel.

How many such circles can be drawn?

[Further Examples on the Construction of Circles will be found on pp. 246, 311.]

PROBLEM 24.

On a given straight line to describe a segment of a circle which shall contain an angle equal to a given angle.

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Let AB be the given st. line, and C the given angle.

It is required to describe on AB a segment of a circle containing an angle equal to C.

Construction.

At A in BA, make the BAD equal to the C. From A draw AG perp. to AD.

Bisect AB at rt. angles by FG, meeting AG in G.

Prob. 2

Now every point in FG is equidistant from A and B ;

... GAGB.

Prob. 14.

With centre G, and radius GA, draw a circle, which must pass through B, and touch AD at A.

Theor. 46.

Then the segment AHB, alternate to the BAD, contains an angle equal to C.

Theor. 49.

NOTE. In the particular case when the given angle is a rt. angle, the segment required will be the semi-circle on AB as diameter. [Theorem 41.]

COROLLARY. To cut off from a given circle a segment containing a given angle, it is enough to draw a tangent to the circle, and from the point of contact to draw a chord making with the tangent an angle equal to the given angle.

It was proved on page 161 that

The locus of the vertices of triangles which stand on the same base and have a given vertical angle, is the arc of the segment standing on this base, and containing an angle equal to the given angle.

The following Problems are derived from this result by the Method of Intersection of Loci [page 93].

EXERCISES.

1. Describe a triangle on a given base having a given vertical angle and having its vertex on a given straight line.

2. Construct a triangle having given the base, the vertical angle, and (i) one other side.

(ii) the altitude.

(iii) the length of the median which bisects the base.

(iv) the foot of the perpendicular from the vertex to the base.

3. Construct a triangle having given the base, the vertical angle, and the point at which the base is cut by the bisector of the vertical angle.

[Let AB be the base, X the given point in it, and K the given angle. On AB describe a segment of a circle containing an angle equal to K; complete the Oce by drawing the arc APB. Bisect the arc APB at P: join PX, and produce it to meet the Oce at C. Then ABC is the required triangle.]

4. Construct a triangle having given the base, the vertical angle, and the sum of the remaining sides.

[Let AB be the given base, K the given angle, and H a line equal to the sum of the sides. On AB describe a segment containing an angle equal to K, also another segment containing an angle equal to half the LK. With centre A, and radius H, describe a circle cutting the arc of the latter segment at X and Y. Join AX (or AY) cutting the arc of the first segment at C. Then ABC is the required triangle.]

5. Construct a triangle having given the base, the vertical angle, and the difference of the remaining sides.

CIRCLES IN RELATION TO RECTILINEAL FIGURES

DEFINITIONS.

1. A Polygon is a rectilineal figure bounded by more than four sides.

A Polygon of five sides is called a Pentagon,

six sides

Hexagon,

2. A Polygon is Regular when all its sides are equal, and all its angles are equal.

3. A rectilineal figure is said to be inscribed in a circle, when all its angular points. are on the circumference of the circle; and a circle is said to be circumscribed about a rectilineal figure, when the circumference of the circle passes through all the angular points of the figure.

4. A circle is said to be inscribed in a rectilineal figure, when the circumference of the circle is touched by each side of the figure; and a rectilineal figure is said to be circumscribed about a circle, when each side of the figure is a tangent to the circle.

Let ABC be the triangle, about which a circle is to be drawn.

Construction. Bisect AB and AC at rt. angles by DS and ES, meeting at S.

Then S is the centre of the required circle.

Prob. 2.

Proof. Now every point in DS is equidistant from A and B;

Prob. 14.

and every point in ES is equidistant from A and C;

.. S is equidistant from A, B, and C.

With centre S, and radius SA describe a circle; this will pass through B and C, and is, therefore, the required circumcircle.

Obs. It will be found that if the given triangle is acuteangled, the centre of the circum-circle falls within it if it is a right-angled triangle, the centre falls on the hypotenuse: if it is an obtuse-angled triangle, the centre falls without the triangle.

NOTE. From page 94 it is seen that if S is joined to the middle point of BC, then the joining line is perpendicular to BC.

Hence the perpendiculars drawn to the sides of a triangle from their middle points are concurrent, the point of intersection being the centre of the circle circumscribed about the triangle.

H.S.G.

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