EXERCISES.

I. In 105° when P passes B where is the LAPB?

2. A, B, C, D are four points on a circle whereof CD is a diameter and E is a point on this diameter. If LAEB=24ACB, E is the centre.

3. The sum of the alternate angles of any octagon in a circle is six right angles.

4. The sum of the alternate angles of any concyclic polygon of 2n sides is 2(1 − 1) right angles.

5. If the angle of a trammel is 60° what arc of a circle will it describe? what if its angle is n°?

6. Trisect a right angle and thence show how to draw a regular 12-sided polygon in a circle.

7. If r, r′ be the radii of two circles, and d the distance between them, the circles touch when d=r±r'.

8. Give the conditions under which two circles have 4, 3, 2, or I common tangent.

9. Prove Ex. 2, 116°, by drawing common tangents to the circles at P, Q, R, and S.

10. A variable chord passes through a fixed point on a circle,

to find the locus of the middle point of the chord. 11. A variable secant passes through a fixed point, to find the locus of the middle point of the chord determined by a fixed circle.

12. In Ex. 11, what is the locus of the middle point of the secant between the fixed point and the circle?

13. In a quadrangle circumscribed to a circle the sums of the

opposite sides are equal in pairs; and if the vertices

be joined to the centre the sums of the opposite angles at the centre are equal in pairs.

14. If a hexagon circumscribe a circle the sum of three alternate sides is equal to that of the remaining three.

15. If two circles are concentric, any chord of the outer which is tangent to the inner is bisected by the point

of contact; and the parts intercepted on any secant between the two circles are equal to one another. 16. If two circles touch one another, any line through the point of contact determines arcs which subtend equal angles in the two circles.

17. If any two lines be drawn through the point of contact of two touching circles, the lines determine arcs whose chords are parallel.

18. If two diameters of two touching circles are parallel, the transverse connectors of their end-points pass through the point of contact.

19. The shortest chord that can be drawn through a given point within a circle is perpendicular to the centre-line through that point.

20. Three circles touch each other externally at A, B, and C.

The chords AB and AC of two of the circles meet the third circle in D) and E. Prove that DE is a diameter of the third circle and parallel to the common centreline of the other two.

21. A line which makes equal angles with one pair of opposite sides of a concyclic quadrangle makes equal angles with the other pair, and also with the diagonals.

22. Two circles touch one another in A and have a common tangent BC. Then BAC is a right angle.

23. OA and OB are perpendicular to one another, and AB is variable in position but of constant length. Find the locus of the middle point of AB.

24. Two equal circles touch one another and each touches one of a pair of perpendicular lines. What is the locus of the point of contact of the circles?

25. Two lines through the common points of two intersecting circles determine on the circles arcs whose chords are parallel.

26. Two circles intersect in A and B, and through B a secant cuts the circles in C and D. Show that LCAD is constant, the direction of the secant being variable.

27. At any point in the circumcircle of a square one of the sides subtends an angle three times as great as that subtended by the opposite side.

28. The three medians of any triangle taken in both length and direction can form a triangle.

SECTION VI.

CONSTRUCTIVE GEOMETRY,

INVOLVING THE PRINCIPLES OF THE FIRST FIVE
SECTIONS, ETC.

117°. Constructive Geometry applies to the determination of geometric elements which shall have specified relations to given elements.

Constructive Geometry is Practical when the determined elements are physical, and it is Theoretic when the elements are supposed to be taken at their limits, and to be geometric in character.

(12°)

Practical Constructive Geometry, or simply "Practical Geometry," is largely used by mechanics, draughtsmen, surveyors, engineers, etc., and to assist them in their work numerous aids known as "Mathematical Instruments" have been devised.

A number of these will be referred to in the sequel.

In "Practical Geometry" the "Rule" (16°) furnishes the means of constructing a line, and the "Compasses " (92°) of constructing a circle.

In Theoretic Constructive Geometry we assume the ability to construct these two elements, and by means of these we are to determine the required elements.

118°. To test the "Rule."

Place the rule on a plane, as at R, and draw a line AB

along its edge. Turn the rule into the position R'. If the edge now coincides with the line

the rule is true.

This test depends upon the pro

R'

perty that two finite points A and B determine one line. (24°, Cor. 2) Def.-A construction proposed is in general called a proposition (2°) and in particular a problem.

A complete problem consists of (1) the statement of what is to be done, (2) the construction, and (3) the proof that the construction furnishes the elements sought.

119°. Problem.-To construct the right bisector of a given line segment.

Let AB be the given segment.

Construction.-With A and B as centres

and with a radius AD greater than half of Ā AB describe circles.

Since AB is < the sum of the radii and > their difference, the circles will meet in two points P and Q.

The line PQ is the right bisector required. Proof.-P and Q are each equidistant from A and B and .. they are on the right bisector of AB; .. PQ is the right bisector of AB.

(54°)

Cor. 1. The same construction determines C, the middle point of AB.

Cor. 2. If C be a given point on a line, and we take A and B on the line so that CA=CB, then the right bisector of the segment AB passes through C and is to the given line.

.. the construction gives the perpendicular to a given line at a given point in the line.

120°. Problem.—To draw a perpendicular to a given line from a point not on the line.

Let L be the given line and P be the point.

Proof.-PDA is the angle in a semicircle,

Cor. Let D be a given point in L. With any centre C and CD as radius describe a circle cutting L again in some point A. Draw the radius ACP, and join D and P. Then DP is to L.

.. the construction draws a 1 to L at a given point in L.

(Compare 119°, Cor. 2)

Cor. 2. Let L be a given line and C a given point.
To draw through C a line parallel to L.

With C as the centre of a circle, construct a figure as given. Bisect PD in E (119°, Cor. 1). Then CE is || to L. For C and E are the middle points of two sides of a triangle of which L is the base.

(84°, Cor. 2)

121°. The Square.—The square consists of two rules with

S

S

B

their edges fixed permanently at right angles, or of a triangular plate of wood or metal having two of its edges at right angles.

To test a square.

Draw a line AB and place the square as at S, so that one edge coincides with the line, and along the other edge draw the line CD.

Next place the square in the position S'. If the edges can