Any point in a line which bisects an angle is equidistant from the sides, 31 ... To ascertain from Sect. 31, the direction and length of a line which will bisect an angle from any given point, with only part of the sides given, 32 Specialties in divisions of or within a circle, 33 ... ... وو A diameter will divide the circle into two equal parts, 34 100 radius, or other right line passing through the centre of the circle, and cutting a chord perpendicularly, will bisect it, 35 Any such line bisecting a chord will cut it perpendicularly, 36 The perpendicular which bisects the chord will bisect the included arc, 37 ... ... ... ... Radius joining the intersecting point of a tangent will be perpendicular to it, 38 Parallels within a circle include equal arcs, 39... 101 Equal chords are equidistant from the centre, and the converse, 40 The exterior and interior angles of an rectilineal figure, are together equal to twice as many right angles as the figure has sides, 41 angles are together equal to four right The interior angles are together equal to twice the number of right angles, less 4, that the figure has sides, 43... To find the side of a square equal in area to the balance in area of several squares, some added and some deducted, 44 In any parallelogram the sum of the squares of the cross diagonals is equal to the sum of the squares of the sides (by reference), 45 Any two sides of a triangle are together longer than the third side, 2 If ... ... any side of a triangle be extended, the exterior angle is greater than either of the interior and opposed angles, 3 108 Any two angles of a triangle are together less than two right angles, 4 ... ... ... ... ... ... ... ... ... 109 ... If one side of a triangle be greater than the second, the In triangles having the same base, and the summits touching a given right line, the sum of the two other sides will be the least where those sides make an equal angle with the given line, 9 111 Although the sides will be less, as in Sect. 9, the area will be greater or less, as the perpendicular height of the triangle is greater or less, 10 ... PAGE 112 113 ... When the base forms a chord and the summits are bounded by an arc, the sides having equal angles at the base and equal lengths will be greatest, 11 Of triangles having one side common, and which cut one another, the sum of the two sides which cut will be greater than that of the two sides which do not cut, 12 114 In a triangle the third side of which passes through a given point, the triangle in which such side is bisected at that point will contain the least area, 13 ... Explanatory remarks on Sects. 15 to 22 following, 14 115 If a right line (D) be divided into any two parts (S and 8), then DxS=Sxs+ S2, and D×s=Sxs+s2. 15 ... ... ... (In this and Sections to 22 the theorems show what is requisite to make equivalents.) ... ... ... If a right line (D) be divided into two parts (S and s), If a right line (D) be divided into two equal parts (R and R), and continued for any extent (C), then (D+C) ×C =(R+C)' R'. 18 ... ... If a right line (D) be divided into any two parts (S and s), then D'+S-2 (D× S) +83, and D'-+s=2 (D× S)+S3. 19 If a right line (D) be divided into any two parts (S and s), then 4 (DxS)=(D+S) -8, and 4 (Dxs)=(D+S)' - S'. 20. (Note that in p. 122 the second 4 (D×S) should be 4 (DX8.)... If a right line (D) be divided into two equal parts (R and R'), and also into two unequal parts (S and s), then S1 +s=R2 2+ (S-R)2 × 2. 21 122 123 ... ... ... If a right line (D) be divided into two equal parts (R and ... ... PAGE 125 ... 127 128 129 Differences in the squares of the sides in an obtuseangled triangle, 24 ... The like generally in triangles, 25, 26 ... ... Explanation that remaining subdivisions of Divisions If two circles cut one another they cannot have the same centre, 30 If two circles touch internally they cannot have the same centre, 31 If any point be taken in the diameter which is not in the centre, variations in lines and distances, 32 ... If from a point without a circle right lines be drawn to the convex (external) and continued to the concave (internal) side, the several difference between external, internal, and total lengths, and tangents, explained, 33 134 The diameter is the greatest chord in any circle, and of all others that which is nearest to the centre is greater than that which is more distant, and the converse, 34 137 If one side of a triangle be bisected, the sum of the squares of the two other sides is double the square of half the side bisected, and of the line from the point of bisection to the opposed angle, 35 138 But in any parallelogram the sum of the squares of the cross diagonals is equal to the sum of the squares of the sides, 36 (Div. C, Sect. 45) DIVISION E. MEASUREMENT OF ANGLES. PAGE 138 Remarks on importance of subject, and objections to rules usually given for measurement, Sect. 1 Description of different kinds of angles, 2 ... ... 140 ... ... Arcs of circles, and division of circumference into degrees, ... ... ... ... Simple illustration of the rule, for proof to a student of its accuracy, 8 Reference to figure giving general application of rule in ... ... Application of rule to right angles with point at summit or circumference; with deduction that all angles in a semicircle are right angles, 15 ..... |