. The Elements of Euclid : viz. the first six books, together with the eleventh and twelfth : the errors, by which Theon, or others, have long ago vitiated these books, are corrected, and some of Euclid's demonstrations are restored : also, the book of Euclid's Data, in like manner corrected. e EA, describe the circle AFG ; fromthe point D draw ^ DF at right angles to EA, and join EBF, AB. b touches the circle BCD. Because E is the centreof the circles BCD, AFG,EA is equal to EF: and ED toEB ; therefore the two sidesAE, EB are equal to the twoFE, ED, and they contain theangle at E comm
. The Elements of Euclid : viz. the first six books, together with the eleventh and twelfth : the errors, by which Theon, or others, have long ago vitiated these books, are corrected, and some of Euclid's demonstrations are restored : also, the book of Euclid's Data, in like manner corrected. e EA, describe the circle AFG ; fromthe point D draw ^ DF at right angles to EA, and join EBF, AB. b touches the circle BCD. Because E is the centreof the circles BCD, AFG,EA is equal to EF: and ED toEB ; therefore the two sidesAE, EB are equal to the twoFE, ED, and they contain theangle at E common to the twotriangles AEB, FED; there-fore the base DF is equal tothe base AB, and the trian-gle FED to the triangle AEB,and the other angles to the other angles « : therefore the angle c is equal to the angle EDF : but EDF is a right angle,wherefore EBA is a right angle : and EB is drawn from the *»>centre : but a straight line diawn from the extremity of adiameter, at right angles to it, touches the circle^: thereforedCor. 16AB touches the circle ; and it is drawn from the given point A. was to be done. But, if the given point be in the circumference of the circle,as the point D, draw DE to the centre E, and DF at right anglesto DE ; DF touches the circle •*.. PROP. XVIII. THEOR. IF a Straight line touches a circle, the straight linedrawn from the centre to the point of contact shallbe perpendicular to the line touching the circle. Let the straight line DE touch the circle ABC in the pointC ; take the centre F, and draw the straight line FC : FC is per-pendicular to DE. For, if it be not, from the point F draw FBG perpendicularto DE ; and because FGC is a right angle, GCF is a an acute a ; and to the greater angle the greatest ^ side is opposite : b 84 THE ELEMENTS Book III. therefore FC is greater than FG ; *^ but FC is equal to FB ; thereforeFB is greater than FG, the lessthan the greater, which is impos-sible ; wherefore FG is n
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