39. A is the extremity of the diameter of a circle, O any point in the diameter. The chord which is bisected at O subtends a greater or less angle at A than any other chord through O, according as O and A are on the same or opposite sides of the centre.

40. Shew that the square on the tangent drawn from any point in the outer of two concentric circles to the inner equals the difference of the squares on the tangents, drawn from any point, without both circles, to the circles.

41. If from a point without a circle, two tangents PT, PT, at right angles to one another, be drawn to touch the circle, and if from T any chord TQ be drawn, and from T' a perpendicular TM be dropped on TQ, then TM=QM.

42. Find the loci:

(1.) of the centres of circles passing through two given points. (2.) Of the middle points of a system of parallel chords in a circle.

(3.) Of points such that the difference of the distances of each from two given straight lines is equal to a given straight line. (4.) of the centres of circles touching a given line in a given point.

(5.) Of the middle points of chords in a circle that pass through a given point.

(6.) Of the centres of circles of given radius which touch a given circle.

(7.) of the middle points of chords of equal length in a circle. (8.) Of the middle points of the straight lines drawn from a given point to meet the circumference of a given circle.

43. If the base and vertical angle of a triangle be given, find the locus of the vertex.

44. A straight line remains parallel to itself while one of its extremities describes a circle. What is the locus of the other extremity ?

45. A ladder slips down between a vertical wall and a horizontal plane : what is the locus of its middle point?

46. AB is the diameter of a circle; ACD is a chord produced to D, so that AC=CD. Find the locus of the point in which BC and the line joining D to the centre intersect.

47. ABC is a line drawn from a point A, without a circle, to meet the circumference in B and C. Tangents are drawn to the circle at B and C which meet in D. What is the locus of Ꭰ ?

48. Two circles intersect in the points A, B; any straight line CDEF is drawn cutting the circles in C, D, E, F ; prove that AC intersects BD and AE intersects BF in points, which lie on a circle passing through A and B.

49. The angular points A, C of a parallelogram ABCD move on two fixed straight lines OA, OC, whose inclination is equal to the angle BCD; shew that the points B, D will move on two fixed straight lines passing through O.

50. On the line AB is described the segment of a circle, in the circumference of which any point is taken. If AC, BC be joined, and a point P taken in AC so that CP is equal to CB, find the locus of P.

51. Find the locus of the centre of the circles circumscribing two trapeziums, into which a parallelogram is divided by any line equal to one of its shorter sides.

52. If a parallelogram be described having the diameter of a given circle for one of its sides, and the intersection of its diagonals on the circumference, shew that the extremity of each of the diagonals moves on the circumference of another circle of double the diameter of the first.

53. One diagonal of a quadrilateral inscribed in a circle is fixed, and the other of constant length. Shew that the sides will meet, if produced, on the circumference of a fixed circle.

We here insert Euclid's proofs of Props. 23, 24 of Book III. first observing that he gives the following definition of similar segments:

DEF. Similar segments of circles are those in which the angles are equal, or which contain equal angles.

PROPOSITION XXIII. THEOREM.

Upon the same straight line, and upon the same side of it, there cannot be two similar segments of circles, not coinciding with each other.

D

If it be possible, on the same base AB, and on the same side of it, let there be two similar segments of Os, ABC, ABD, which do not coincide.

Because ADB cuts o ACB in pts. A and B, they cannot cut one another in any other pt., and.. one of the segments must fall within the other.

Let ADB fall within ACB.

Draw the st. line BDC and join CA, DA.

Then segment ADB is similar to segment ACB,

.. LADB=L ACB.

A =

Or the extr. 4 of a ▲ the intr. and opposite, which is impossible;

.. the segments cannot but coincide.

Q. E. D.

PROPOSITION XXIV. THEOREM.

Similar segments of circles, upon equal straight lines, are equal to one another.

Let ABC, DEF be similar segments of Os on equal st. lines AB, DE.

Then must segment ABC=segment DEF.

For if segment ABC be applied to segment DEF, so that A may be on C and AB on DE, then B will coincide with E, and AB with DE ;

.. segment ABC must also coincide with segment DEF;

.. segment ABC=segment DEF.

III. 23.

Ax. 8.

Q. E. D.

We gave one Proposition, C, page 150, as an example of the way in which the conceptions of Flat and Reflex Angles may be employed to extend and simplify Euclid's proofs. We here give the proofs, based on the same conceptions, of the important propositions XXII. and XXXI.

PROPOSITION XXII. THEOREM.

The opposite angles of any quadrilateral figure, inscribed in a circle, are together equal to two right angles.

Let ABCD be a quadrilateral fig. inscribed in a C.

Then must each pair of its opposite s be together equal to two rt. 4 s.

.. sum of 4 s at 0=twice sum of 2 s BAD, BCD.

But sum of 4s at 0=4 right s;

I. 15, Cor. 2.

..twice sum of 4 s BAD, BCD=4 right 4 s;

.. sum of 48 BAD, BCD=two rights.

Similarly, it may be shewn that

sum of 48 ABC, ADC=two right 4 s.

Q. E. D.