37. To construct a triangle, of which the base, the difference of the other two sides, and the vertical angle are given.

38. To construct a triangle, of which the base, the altitude, and the vertical angle are given.

39. To construct a triangle, of which the perimeter, the altitude, and the vertical angle are given.

40. To find a point from which the three straight lines drawn to three given points shall make equal angles with each other. 41. In the circumference of a circle to find a point from which a given straight line shall subtend an angle equal to a given angle, whenever it is possible.

XXXV.

42. If two chords in a circle intersect each other at right angles, the sum of the squares of their four segments is equal to the square of the diameter.

43. In a circle, the rectangle by the segments of a chord is equal to the difference of the squares of the radius, and of the straight line joining the centre with the point of section.

44. In a circle, if a perpendicular be drawn from any point in a chord to a diameter, the rectangle by the segments of the diameter is equal to the rectangle by the segments of the chord, together with the square of the perpendicular.

45. If two tangents be drawn at the extremities of the diameter of a circle, and a third tangent be drawn at any other point, to meet them, the rectangle by its segments between the two other tangents and the point of contact is equal to the square described on the radius.

XXXVI.

46. If a given circle be cut by any number of circles which all pass through two given points without it, the straight lines which join the points of intersection are either parallel, or all meet, if produced, in the same point.

47. If two circles touch one another externally, and a common tangent be drawn, not meeting both at the same point, the square of the part of this line intercepted between the points of contact is equal to the rectangle contained by the diameters of the circles.

XXXVII.

48. If three circles touch one another externally, the tangents at the points of contact meet all in one point.

49. To describe a circle which shall pass through two given points, and touch a given straight line.

50. To describe a circle which shall have its centre in a given straight line, shall pass through a given point, and touch a given straight line.

51. To describe a circle which shall touch each of two given straight lines and a given circle.

7.13

BOOK IV.

PROP. III.

1. An equilateral triangle inscribed in a circle is a fourth part of an equilateral triangle described about the same circle.

2. If a triangle be described about a given circle, the rectangle by the perimeter of the triangle and the radius of the circle is double the area of the triangle.

IV.

d. 3. To describe a circle which shall touch the base of a given triangle and the other sides produced.

4. If a circle be inscribed in a triangle, the distance of any angle of the triangle from the point of contact of the circle with one of the sides which contain it, is equal to half the excess of the sum of these sides above the side opposite to the angle.

5. The square of the side of an equilateral triangle inscribed in a circle is triple the square of the side of a hexagon inscribed in the same circle.

6. The diameter of the circle inscribed in a right-angled triangle is equal to the excess of the sum of the sides which contain the right angle above the hypotenuse.

7. A circle may be inscribed in any quadrilateral figure, provided the sums of its opposite sides are equal.

V.

8. The diameter of the circle described about an equilateral triangle is double the diameter of the circle inscribed in the same triangle.

9. The angle contained by two straight lines drawn from either of the angular points of a triangle to the centres of its inscribed and circumscribing circles is half the difference between the other angles of the triangle.

VI.

10. In a given circle to inscribe a rectangle equal to a given rectilinear figure.

VII.

11. In a given circle to inscribe four equal circles, mutually touching each other, and the given circle.

X.

12. The base of the triangle described in Prop. X. is the side of a regular decagon inscribed in the larger circle, and of a regular pentagon inscribed in the smaller circle.

13. In an isosceles triangle which has each of the angles at the base double the third angle, the difference of the squares of one side and the base is equal to their rectangle.

XI.

14. Upon a given straight line to describe an equilateral and equiangular pentagon.

XV.

15. Upon a given straight line to describe an equilateral and equiangular hexagon.

BOOK V.

PROP. XVI.

1. If the first of four magnitudes of the same kind have to the second a greater ratio than the third has to the fourth, the first shall have to the third a greater ratio than the second has to the fourth.

XVII. and XVIII.

2. If four magnitudes of the same kind be proportionals, of which the first is the greatest, the sum of the two extremes is greater than the sum of the two means.

3. If of four magnitudes, the first, together with the second, have to the second a greater ratio than the third, together with the fourth, has to the fourth, the first shall have to the second a greater ratio than the third has to the fourth.

4. If the first have to the second a greater ratio than the third has to the fourth, the first, together with the second, shall have to the second a greater ratio than the third, together with the fourth, has to the fourth.

5. If the first have to the second a greater ratio than the third has to the fourth, the first, together with the third, shall have to the second, together with the fourth, a greater ratio than the third has to the fourth, but a less ratio than the first has to the second. 6. If there be three magnitudes of the same kind, of which the first is less than the second, the first, together with the third, shall have to the second, together with the third, a greater ratio than the first has to the second.

7. If the first, together with the second, have to the second a greater ratio than the third, together with the fourth, has to the fourth, then shall the first, together with the second, have to the first a less ratio than the third, together with the fourth, has to the third.

XXII.

8. If the first have to the second the same ratio which the fourth has to the fifth, but the second to the third a greater ratio than the fifth has to the sixth, the first shall have to the third a greater ratio than the fourth has to the sixth.

XXIII.

9. If the first have to the second the same ratio which the fifth has to the sixth, but the second to the third a greater ratio than the fourth has to the fifth, the first shall have to the third a greater ratio than the fourth has to the sixth.

BOOK VI.

PROP. I.

1. Triangles and parallelograms which have equal bases, are to one another as their altitudes.

II.

2. The straight lines which join the extremities of parallel radii of two unequal circles, when produced, pass all through the same point.

3. From a given point to draw a straight line which shall cut off from the two lines which contain a given rectilineal angle, parts which have a given ratio to one another.

TTT.

4440

4. If a straight line be divided in a given point, to construct upon it a triangle having a given vertical angle, and its other sides in the same proportion as the segments of the base.

III. A. and IV.

5. If one angle at the base of a triangle be double of the other, the less side is equal to the sum or difference of the segments of the base made by the perpendicular from the vertex, according as the angle is greater or less than a right angle.

IV.

6. The diameter of a circle is a mean proportional between the sides of an equilateral triangle and hexagon described about the circle. 7. If two triangles have an angle of the one equal to an angle of the other, the triangles are to one another as the rectangle by the sides about those angles respectively.

8. A straight line drawn from the vertex of a triangle to the base, cuts all straight lines which are parallel to the base, and terminated by the other sides of the triangle in the same ratio as the segments of the base.

9. If from a point without a circle there be drawn two straight lines, the one touching and the other cutting the circle, and chords be drawn from the point of contact to the two points of section, the whole of the cutting line has to the part of it without the circle the duplicate ratio of the greater chord to the less.

10. If one side of a triangle be produced and another shortened by the same quantity, the line which joins the points of section will be divided by the base, in the inverse ratio of the sides.

X.

11. To divide a given circular arc into two such parts that the chords of its segments shall have a given ratio.

XV.

12. To describe an isosceles triangle which shall be equal to a given triangle, and have one of its angles equal to an angle of the given triangle.

XVI.

13. Double the area of a triangle is to the rectangle contained by any two of its sides as the third side to the diameter of the circumscribing circle.

D.

14. If ABCD be any parallelogram, and if a circle be described passing through the point A, and cutting the sides AB, AC, and the diagonal AD, in the points F, G, H respectively; then the rectangle AD.AH is equal to the sum of the rectangles AB.AF and AC.AG.

MISCELLANEOUS.

1. If a straight line be drawn from an angle of a scalene triangle to the point of bisection of the base, the distance of any point in that line from the greater of the other two angles of the triangle is less than its distance from the less; and the difference between the two distances is less than the difference between the two sides of the triangle which are opposite to those angles. 2. If a straight line be drawn from an angle of a scalene triangle, at right angles to the opposite side, the distance of any point in that line from the greater of the other two angles of the triangle is less than its distance from the less; and the difference between the two distances is greater than the difference between the two sides of the triangle which are opposite to those angles. 3. The square of the side of a regular pentagon inscribed in a given circle, is equal to the square of the side of a regular decagon, together with the square of the side of a regular hexagon both inscribed in the circle.

4. If in the figure of Prop. 2, Book VI., BE and DC intersect one another in F, and AF be joined and produced, this line will bisect both BC and DE.