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The Contents of the Fifth and Sixth Books of Euclid (with a Note on ...
Euclid,Micaiah John Muller Hill
No preview available - 2015
ABCD alternatives angles base Book called centre circle compartments consequence construct contained corresponding sides definition described determine difference divided draw drawn duplicate ratio ENUNCIATION equal equal angles equimultiples Euclid's EXAMPLE exist expressed by Fig Fifth figures follows four fourth given greater harmonic Hence idea integer internally Join kind known length magnitudes manner mean proportional measure necessary Note opposite pair parallel parallel to BC parallelogram perpendicular placed possible produced proof Prop PROPOSITION prove ratio rect rectangle rectilineal figures relative multiple scale result right angle second column sector segments similar similarly square stage standing straight line Take taken term third three magnitudes triangle ABC triangles are similar vertex
Page 99 - IF two triangles have one angle of the one equal to one angle of the other, and the sides about the equal angles proportionals : the triangles shall be equiangular, and shall have those angles equal which are opposite to the homologous sides.
Page xviii - Wherefore, in equal circles &c. QED PROPOSITION B. THEOREM If the vertical angle of a triangle be bisected by a straight line which likewise cuts the base, the rectangle contained by the sides of the triangle is equal to the rectangle contained by the segments of the base, together with the square on the straight line which bisects the angle.
Page 99 - Prove that similar triangles are to one another in the duplicate ratio of their homologous sides.
Page xvi - If the vertical angle of a triangle be bisected by a straight line which also cuts the base, the segments of the base shall have the same ratio which the other sides of the triangle have to one another...
Page 99 - ABC, DEF have one angle in the one equal to one angle in the other, viz. the angle BAC to the angle EDF, and the sides about...
Page xvi - In a right triangle, if a perpendicular is drawn from the vertex of the right angle to the hypotenuse : I.
Page 35 - The first of four magnitudes is said to have the same ratio to the second, which the third has to the fourth, when any equimultiples whatsoever of the first and third being taken, and any equimultiples whatsoever of the second and fourth ; if the multiple of the first be less than that of the second, the multiple of the third is also less than that of the fourth: or, if the multiple of the first be equal to that of the second, the multiple of the third is also equal to that of the fourth : or, if...
Page 84 - If they do not intersect, show that the radical axis is perpendicular to the line joining the centres of the circles...
Page 100 - If an angle of a triangle be bisected by a straight line, which likewise cuts the base; the rectangle contained by the sides of the triangle is equal to the rectangle contained by the segments of the base, together with the square of the straight line bisecting the angle.