. Elements of geometry : containing books I to III. s ? the 0 £0, ££.Then v E=BE, .: l 0BE= l BOE ; I. a. .-. z AEB=tmcei OBfi ; I. 32. and v AE BE, .-. - .17;/;= ^BAE: 1. \..-. OEB«twice i ABE ; I. 32. .•. sum of a I/.7., OEB- twice stun ofzs 0£i?, ABE,that is, two righl angles=twicer 02L4 ; .. /..I is a right angle,and .. AH is a tangent to the • BCUi ill. it may be Bhewn that .1/ is a tangent to © , l be described about a that the sum of AB equal to the sum of A I) and BC. Book m. xvm. 145 Proposition XVIII. Theorem: If a straight lint
. Elements of geometry : containing books I to III. s ? the 0 £0, ££.Then v E=BE, .: l 0BE= l BOE ; I. a. .-. z AEB=tmcei OBfi ; I. 32. and v AE BE, .-. - .17;/;= ^BAE: 1. \..-. OEB«twice i ABE ; I. 32. .•. sum of a I/.7., OEB- twice stun ofzs 0£i?, ABE,that is, two righl angles=twicer 02L4 ; .. /..I is a right angle,and .. AH is a tangent to the • BCUi ill. it may be Bhewn that .1/ is a tangent to © , l be described about a that the sum of AB equal to the sum of A I) and BC. Book m. xvm. 145 Proposition XVIII. Theorem: If a straight lint touch a circle, the straight lim drawn fromtin centrt to ttu point of contact must be perpendicular to the line touching the ?. Let the st. line DE touch the S ABC in the pt. 0 the centre, and join OC Then OC be ± to DE. For if it be not, draw OBFi. to DE, meeting the Oce in B. Then Vl OFCisu it. angle, /. l OCF is less than a it. angle, I. 17. and .-. OC is greater than OF. I. 19. But OC=OB, .. OB is greater than OF, which is impossible ; .. OF is not _l to DE, and in the same way it may In-shewn that no other line drawn from 0, but OC, is _L to DE , .: 00 is _ to DE. Q. E. U. Ex. If two sl aight lines intersect, the centres of all circlestouched by buth lines lie in two lines at right angles to each other. Note. Prop. xvm. might be stated thus :—AU radiicircle are normals to the circle at the points where tluij meet thecircumference, r46 EUCLIDS ELEMENTS. [Book III. 3IT10N X IX. Till If a straight liii touch circle, and from (I, ptact i straight line be drawn at right angles to the fouchingline, the centre C-f th( circle must In in that
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