Elements of geometry and trigonometry . ys bemade to pass through three given points, not in the samestraight line : we say farther, that but one can be describedthrough them. For, if there were a second circumference passing through tliethree given points A, B, C, its centre could not be out of theline DE, for then it would be unequally distant from A and B(Book I. Prop. XVI.); neither could it be out of the line FG, fora like reason ; therefore, it would be in both the lines DE, two straight lines cannot cut each other in more than onepoint ; hence there is but one circumference which
Elements of geometry and trigonometry . ys bemade to pass through three given points, not in the samestraight line : we say farther, that but one can be describedthrough them. For, if there were a second circumference passing through tliethree given points A, B, C, its centre could not be out of theline DE, for then it would be unequally distant from A and B(Book I. Prop. XVI.); neither could it be out of the line FG, fora like reason ; therefore, it would be in both the lines DE, two straight lines cannot cut each other in more than onepoint ; hence there is but one circumference which can passthrough three given points. Cor. Two circumferences cannot meet in more than twoponits ; for, if they have three common points, there would betwo circumferences passing through the same three points ;which has been shown by the proposition to be impossible. BOOK III. 47 PROPOSITION VIII. THEOREM. Two equal chords are equally distant from the centre ; and of twounequal chords, the less is at the greater distance from First. Suppose the chord AB =DE. Bisect these chords by the per-pendiculars CF, CG, and draw thera(hi CA, CD. In the riCI ; hence of two unequalchords, the less is the farther from the centre. PROPOSITION IX. THEOREM. A straight line perpendicular to a radius, at its e
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