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10. The part of a tangent to a circle, intercepted between two tangents drawn from the extremities of any diameter, subtends a right angle at the centre; and is divided at the point of contact, so that the radius is a mean proportional between the segments.
11. If through the bisection of the base of a triangle any line be drawn, cutting one side of the triangle, and the other side produced, and a line be drawn through the vertex parallel to the base, the former line will be cut harmonically.
12. If one side of a right-angled triangle be double of the other, prove that a perpendicular from the right angle on the hypotenuse will divide it into segments in the ratio of 4 to 1.
13. From a given angle, to cut off a triangle equal to a given
14. To cut off from a given triangle another similar to it, and in a given ratio to it.
15. Given the base, the altitude, and the ratio of the sides of a triangle, to construct it.
16. If from the extremities of the base of a triangle, lines be drawn bisecting the opposite sides, they will divide each other in the ratio of 2 to 1.
17. If a line be drawn parallel to the base of a triangle to meet the sides, and the alternate extremities of this line and of the base be joined, the line drawn from the vertex through the intersection of the connecting lines will bisect the base, and will be cut harmonically.
18. The inclination of two chords of a circle is measured by half the sum, or half the difference of the intercepted arcs,' according as they intersect internally or externally.
This property of the circle suggested a considerable improvement in the form of astronomical angular instruments.
19. If three lines be in continued proportion, the first is to the third as the square on the difference between the first and second, to the square on the difference between the second and third.
20. If a line bisect the angle adjacent to the vertical angle of a triangle, and meet the base produced, the difference between the square on that line and the rectangle under the external segments of the base, is equal to the rectangle under the sides of the triangle.
This may be considered as another case of VI. B.
21. If two tangents and a secant be drawn to a circle from a
point without it, and the points of contact be joined by a straight line, the secant will be cut harmonically.
22. If two triangles have two angles together equal to two right angles, and other two angles equal, the sides about their remaining angles are proportional.
9:23. If a line be drawn through any point in the base of an isosceles triangle, so as to cut off from one side and add to the other equal segments, it will be bisected by the base.
24. If from the angular points of a triangle, lines be drawn through any point within it to meet the opposite sides, and if from the point of section of the base, lines be drawn through the other two points of section to meet a line drawn through the vertex parallel to the base, the intercepted portion of the latter is bisected in the vertex.
25. If from the extremities of the base of a triangle, lines be drawn through any point in the perpendicular to meet the sides, lines joining the points of section of the sides with that of the base, will make equal angles with the base.
26. Harmonicals cut all straight lines that intersect them harmonically.
27. If lines be drawn from the angular points of a triangle, through any point within it, to meet the opposite sides, and the points of section be joined, the former lines will be cut harmonically; and if these lines be produced to meet the sides produced, the latter will be cut harmonically.
28. If through any point within or without a triangle, lines be drawn from the angular points to meet the opposite sides, and if lines joining the points of section be produced to meet the sides produced if necessary, the three points of concourse are in one line.
29. To draw a straight line, so that the part of it intercepted between one side of a given isosceles triangle and the other side produced, shall be equal to a given line, and be bisected by the base.
30. Given the altitude, the vertical angle, and the sum or difference of the sides of a triangle, to construct it.
31. Given the base, the altitude, and the sum or difference of the sides of a triangle, to construct it.
32. Given the altitude of a triangle, the difference of the angles at the base, and the sum or difference of the sides, to construct it.
33. The altitude, the difference of the segments of the base, and either the sum or difference of the two sides of a triangle, are given to construct it.
34. Given the altitude, the vertical angle, and the perimeter of a triangle, to construct it.
35. If perpendiculars be drawn from the extremities of the base of a triangle on a straight line that bisects the angle opposite to the base, the area of the triangle is equal to the rectangle contained by either of the perpendiculars, and the segment of the bisecting line between the angle and the other perpendicular.
36. If perpendiculars be drawn from the extremities of the base of a triangle on a straight line which bisects the angle opposite to the base; four times the rectangle contained by the perpendiculars is equal to the rectangle contained by two straight lines, one of which is the base increased by the difference of the sides, and the other the base diminished by the difference of the sides.
37. If perpendiculars be drawn from the extremities of the base of a triangle on a straight line which bisects the angle opposite to the base; four times the rectangle contained by the segments of the bisecting line between the angle and the perpendiculars, is equal to the rectangle contained by two straight lines, one of which is the sum of the sides increased by the base, and the other the sum of the sides diminished by the base.
38. Apply the three exercises (35, 36, and 37) to prove proposition L of the Sixth Book.
ON THE QUADRATURE OF THE CIRCLE, AND THE RECTIFICATION OF ITS CIRCUMFERENCE.
1. The determination of a square equal to a given surface, is called the quadrature of that surface.
2. The determination of a straight line equal to a curve line, is called the rectification of that curve line.
3. A mixed line is composed of straight and curve lines.
4. A mixtilineal space is a space contained by a mixed line.
5. The inclination of a straight line and a curve is the angle contained by the former, and a tangent to the latter at the point of intersection.
6. The supplemental chord of an arc is the chord of its defect from a semicircumference.
1. Of all lines that can join two points, there must be at least one such that no other is less than it; and if there be only one such, it is the least.
2. If any number of lines fulfil certain conditions, and if a line fulfilling the same conditions can always be found less than any of them, except one, this one must be the least.
PROPOSITION I. THEOREM.
If two rectilineal figures be on the same side of the same base, and if one of them be wholly encompassed by the other, and be also concave internally, the sum of its sides is less than that of the other.
Given the figures ACDB, AEFB upon the same base AB; to AE + EF + FB7 AC + CD + DB.
(Const.) Produce AC to G; K. (Dem.) Then (I. 20) AG, CG + GFCF,
join C, F, and produce CD to
AE + EG >
CF + FK
CK, and DK + KBDB.
which is AC +.
CD + DB, still more then is AE + EF
And in a similar manner the proposition is proved for polygons.
Of all the lines, straight and curve, that can join two points, the straight line is the least.
Let A, B, be any two points, the straight line AB is the shortest line that can join them.
1. Let the curve line ADB, concave towards AB, join them. Then take any point C in AB, and from the centres A and B, with the radii AC and CB respectively, describe the arcs CD and CE, passing through C, and cutting ADB in D and E. (Dem.) Then, since the arcs touch at C (III. 12), and cannot touch at any other point (III. 13), the point E must be without the arc CD, without the arc CE; and the line DE between them must be of some length. Now, if the points D and E be made to coincide with C, A and B remaining fixed, the curve lines AD and EB would connect A and B; and hence a line shorter than ADEB can connect these points. The same can be proved of any other concave line, therefore the straight line AB is
shorter than any of them (Ax. 1).
2. Let a line ACDB, partly concave and partly convex towards AB, connect the points. Let AC and
DB be concave, and CD the convex portion. Draw the straight lines AC, CD, and DB. Then (by 1st case) the straight lines AC, CD, and DB are less than the curve lines AC; CD, and DB respectively;
and the sum of the former is therefore less than the whole curve line; hence a line shorter than it has been found.
3. Let the crooked line ACDB join A and B. be joined by a straight line, AC + CD > AD,
Then, if AD and AD + DB