PROPOSITION X. THEOREM. Two parallels intercept equal arcs of a circumference. There may be three cases: both parallels may be secants; one may be a secant and the other a tangent; or, both may be tangents. 1o. Let the secants AB and DE be parallel then will the intercepted arcs MN and PQ be equal. For, draw the radius CH perpendicular to the chord MP; it will also be perpendicular to NQ (B. I., P. XX., C. 1), and I will be at the middle point of the arc MHP, and also of the arc NHQ hence, MN, which is the difference of HN and HM, D H A -B M P N -E is equal to PQ, which is the difference of HQ and HP (A. 3); which was to be proved. 2o. Let the secant AB and tangent DE, be parallel : then will the intercepted arcs MH and PHI be equal. · For, draw the radius CHI 3°. Let the tangents DE and IL be parallel, and let H and K be their points of contact: then will the intercepted arcs HMK and HPK be equal. For, draw the secant AB parallel to DE; then, from what has just been shown, we shall have HM equal to HP, and MK equal to PK: hence, HMK, which is the sum of HM and MK, is equal to HPK, which is the sum of HP and PK; which was to be proved. D A -E -B M P I -L K PROPOSITION XI. THEOREM. If two circumferences intersect each other, the points of in tersection will be in a perpendicular to the line joining their centres, and at equal distances from it. Let the circumferences, whose centres are C and D, intersect at the points A and B: then will CD be perpendicular to AB, and AF will be equal to BF For, the points A and B, being on the circumference whose centre is C, are equally distant from C; and being on C F D B the circumference whose centre is D, they are equally distant from D: hence, CD is perpendicular to AB at its middle point (B. I., P. XVI., C.); which was to be proved. PROPOSITION XII. THEOREM. If too circumferences intersect each other, the distance be tween their centres will be less than the sum, and greater than the difference, of their radii. Let the circumferences, whose centres are C and D, intersect at A: then will CD be less than the sum, and greater than the difference of the radii of the two circles. For, draw AC and AD, forming the triangle ACD. Then will CD be less than the sum of AC and AD, C D and greater than their difference (B. I., P. VII.); which was to be proved. If the distance between the centres of two circles is equal to the sum of their radii, they will be tangent externally. Let C and D be the centres of two circles, and let the distance between the centres be equal to the sum of the radii then will the circles be tangent externally. For, they will have a point A, on the line CD, common, and they will have no other point in common; for, if they had two points in common, the distance between their centres would be less than the sum of E C D their radii; which is contrary to the hypothesis: hence, they are tangent externally; which was to be proved. If the distance between the centres of two circles is equal to the difference of their radii, one will be tangent to the other internally. Let C and D be the centres of two circles, and let the distance between these centres be equal to the difference of the radii: then will the one be tangent to the other internally. For, they will have a point A, on DC, E C D hence, one touches the other internally; which was to be proved. Cor. 1. If two circles are tangent, either externally or internally, the point of contact will be on the straight line drawn through their centres. Cor. 2. All circles whose centres are on the same straight line, and which pass through a common point of that line, are tangent to each other at that point. And if a straight line be drawn tangent to one of the circles at their common point, it will be tangent to them all at that point. Scholium. From the preceding propositions, we infer that two circles may have any one of six positions with respect to each other, depending upon the distance between their centres: 1o. When the distance between their centres is greater than the sum of their radii, they are external, one to the other: 2°. When this distance is equal to the sum of the radii, they are tangent, externally: 3o. When this distance is less than the sum, and greater than the difference of the radii, they intersect each other: 4°. When this distance is equal to the difference of their radii, one is tangent to the other, internally: 5o. When this distance is less than the difference of the radii, one is wholly within the other: 6o. When this distance is equal to zero, they have a common centre; or, they are concentric. PROPOSITION XV. THEOREM. In equal circles, radii making equal angles at the centre, intercept equal arcs of the circumference; conversely, radii which intercept equal arcs, make equal angles at the centre. 1o. In the equal circles ADB and EGF, let the angles ACD and EOG be equal: then will the arcs AMD and ENG be equal. B M F E N For, draw the chords AD and EG; then will the triangles ACD and EOG have two sides and their included angle, in the one, equal to two sides and their included angle, in the other, each to each. in all their parts; consequently, But, if the chords AD and EG and ENG are also equal (P. IV.); which was to be proved. They are, therefore, equal AD is equal to EG. are equal, the arcs AMD |