drawn parallel to the other, intersecting the adjacent side of the

trapezium, and a second line to the extremity of that other inter-

secting the circumference: the line joining the two points of inter-

section will pass through the same point.

65. If the diagonals of a quadrilateral figure inscribed in a circle,

cut each other at right angles; the rectangles contained by the oppo-

site sides are together double of the quadrilateral figure.

66. If a rectangular parallelogram be inscribed in a right-angled

triangle, and they have the right angle common; the rectangle con-

tained by the segments of the hypothenuse is equal to the sum of the

rectangles contained by the segments of the sides about the right

angle.

67. If on the diameter of a semicircle two equal circles be de-

scribed, and in the curvilinear space included by the three circum-

ferences a circle be inscribed; its diameter will be to that of the

equal circles in the proportion of two to three.

68. If through the middle point of any chord of a circle two

chords be drawn; the lines joining their extremities will intersect

the first chord at equal distances from the middle point.

69. The longest side of a trapezium being given, and made the

diameter of the circumscribed circle; also the distance between its

extremity and the intersection of the opposite side produced to meet

it, and the angle formed by the intersection of the diagonals: to

construct the trapezium.

70. The diagonals of a quadrilateral figure inscribed in a circle

are to one another as the sums of the rectangles of the sides which

meet their extremities.

71. The square described on the side of an equilateral and equi-

angular pentagon inscribed in a circle, is equal to the sum of the

squares of the side of a regular hexagon and decagon inscribed in the

same circle.

72. If the opposite sides of an irregular hexagon inscribed in a

circle be produced till they meet; the three points of intersection

will be in the same straight line.

SECTION VII. Page 232.

1. THE vertical angle of an oblique-angled triangle inscribed in

a circle, is greater or less than a right angle, by the angle contained

by the base and the diameter drawn from the extremity of the base.

2. If from the vertex of an isosceles triangle a circle be described

with a radius less than one of the equal sides, but greater than the

perpendicular; the parts of the base cut off by it will be equal.

3. If a circle be inscribed in a right-angled triangle; the differ-

ence between the two sides containing the right angle and the

hypothenuse, is equal to the diameter of the circle.

4. If a semicircle be inscribed in a right-angled triangle so as to

touch the hypothenuse and perpendicular, and from the extremity of

its diameter a line be drawn through the point of contact, to meet

the perpendicular produced; the part produced will be equal to the

perpendicular.

5. If the base of any triangle be bisected by the diameter of its

circumscribing circle, and from the extremity of that diameter a per-

pendicular be let fall upon the longer side; it will divide that side

into segments, one of which will be equal to half the sum, and the

other to half the difference of the sides.

6. The same supposition being made as in the last proposition;

if from the point where the perpendicular meets the longer side,

another perpendicular be let fall on the line bisecting the vertical

angle; it will pass through the middle of the base.

7. If a point be taken without a circle, and from it tangents be

drawn to the circle, and another point be taken in the circumference

between the two tangents, and a tangent be drawn to it; the sum of

the sides of the triangle thus formed is equal to the sum of the two

tangents.

8. Of all triangles on the same base and between the same

parallels, the isosceles has the greatest vertical angle.

COR. Of all triangles on the same base,. and having the same

vertical angle, the isosceles is the greatest.

9. If through the vertex of an equilateral triangle a perpendicular

be drawn to the side, meeting a perpendicular to the base drawn

from its extremity; the line intercepted between the vertex and the

latter perpendicular is equal to the radius of the circumscribing

circle.

10. If a triangle be inscribed in a semicircle, and a perpen-

dicular drawn from any point in the diameter meeting one side, the

circumference, and the other side produced; the segments cut off

will be in continued proportion.

11. If a triangle be inscribed in a semicircle, and one side be

equal to the semidiameter; the other side will be a mean proportional

between that side and a line equal to that side and the diameter

together.

12. If a circle be inscribed in a right-angled triangle; to deter-

mine the least angle that can be formed by two lines drawn from the

extremity of the hypothenuse to the circumference of the circle.

13. If an equilateral triangle be inscribed in a circle, and through

the angular points another be circumscribed; to determine the ratio

which they bear to each other.

14. A straight line drawn from the vertex of an equilateral

triangle inscribed in a circle to any point in the opposite circum-

ference, is equal to the two lines together, which are drawn from

the extremities of the base to the same point.

15. If the base of a triangle be produced both ways, so that each

part produced may be equal to the adjacent side, and through the

extremities of the parts produced and the vertex a circle be de-

scribed; the line joining its centre and the vertex of the triangle

will bisect the angle at the vertex.

16. If an isosceles triangle be inscribed in a circle, and from the

vertical angle a line be drawn meeting the circumference and the

base; either equal side is a mean proportional between the segments

of the line thus drawn.

17. If from the extremities of one of the equal sides of an

isosceles triangle inscribed in a circle, tangents be drawn to the circle

and produced to meet; two lines drawn to any point in the circum-

ference from the point of concourse and one point of contact, will

divide the base (produced if necessary) in geometrical proportion.

18. If on the sides of a triangle segments of circles be described

similar to a segment on the base, and from the extremities of the

base tangents be drawn intersecting their circumferences; the points

of intersection and the vertex of the triangle will be in the same

straight line.

19. The centre of the circle which will touch two semicircles

described on the sides of a right-angled triangle is in the middle

point of the hypothenuse.

COR. Its diameter will be equal to the sides together.

20. If on the three sides of a right-angled triangle semicircles be

described, and with the centres of those described on the sides,

circles be described touching that described on the base; they will

also touch the other semicircles.

21. If from any point in the circumference of a circle perpen-

diculars be drawn to the sides of the inscribed triangle; the three

points of intersection will be in the same straight line.

22. The base of a right-angled triangle not being greater than

the perpendicular; if on any line drawn from the vertex to the base

a semicircle be described, and a chord equal to the perpendicular

placed in it, and bisected; the point of bisection will always fall

within the triangle.

23. The straight line bisecting any angle of a triangle inscribed

in a given circle, cuts the circumference in a point, which is equi-

distant from the extremities of the side opposite to the bisected

angle, and from the centre of a circle inscribed in the triangle.

24. The perpendicular from the vertex on the base of an equi-

lateral triangle is equal to the side of an equilateral triangle inscribed

in a circle whose diameter is the base.

25. If an equilateral triangle be inscribed in a circle, and the

adjacent arcs cut off by two of its sides be bisected; the line joining

the points of bisection will be trisected by the sides.

26. If any triangle be inscribed in a circle, and from the vertex

a line be drawn parallel to a tangent at either extremity of the base;

this line will be a fourth proportional to the base and two sides.

27. If a triangle be inscribed in a circle, and from its vertex

lines be drawn parallel to tangents at the extremities of its base; they

will cut off similar triangles.

COR. 1. The rectangle contained by the segments of the base

adjacent to the angles is equal to the square of either line drawn

from the vertex.

COR. 2. Those segments are also in the duplicate ratio of the

adjacent sides.

28. If one circle be circumscribed and another inscribed in a

given triangle, and a line be drawn from the vertical angle to the

centre of the inner, and produced to the circumference of the outer

circle; the whole line thus produced has to the part produced the

same ratio that the sum of the sides of the triangle has to the base.

29. If in a right-angled triangle, a perpendicular be drawn from

the right angle to the hypothenuse, and circles inscribed within the

triangles on each side of it; their diameters will be to each other as

the subtending sides of the right-angled triangle.

30. To find the locus of the vertex of a triangle, whose base and

ratio of the other two sides are given.

31. A given straight line being divided into any three parts; to

determine a point such, that lines drawn to the points of section and

to the extremities of the line shall contain three equal angles.

32. If two equal lines touch two unequal circles, and from the

extremities of them lines containing equal angles be drawn cutting

the circles, and the points of section joined; the triangles so formed

will be reciprocally proportional.

33. If from an angle of a triangle a line be drawn to cut the

opposite side, so that the rectangle contained by the sides including

the angle be equal to the rectangle contained by the segments of the

side together with the square of the line so drawn; that line bisects

the angle.

34. In any triangle, if perpendiculars be drawn from the angles

to the opposite sides, they will all meet in a point.

35. If from the extremities of the base of any triangle, two per-

pendiculars be let fall on the line bisecting the vertical angle; and

through the points where they meet that line, and the point in the

base, whereon the perpendicular from the vertical angle falls, a circle.

be described; that circle will bisect the base of the triangle.

36. If from one of the angles of a triangle a straight line be

f