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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 ABGD arc CD Archimedes BEFG bisected BLNO centre congruent corresponding sides cross-ratio diagonal duplicate ratio EFGH Enunciation EQPS equal angles equimultiples Euclid's EXAMPLE express the facts expressed by Fig Fifth Book follows four harmonic points four magnitudes given straight line greater Hence the scale Hence the triangles hypotenuse i.e. the fact kind mean proportional middle point parallel straight lines parallel to BC parallelogram point of division PQRST proof PROPOSITION radical axis ratio compounded ratio of equality rect rectangle contained relative multiple scale required to prove respectively equal right angle second column side corresponding similar figures similar triangles similarly described Sixth Book square supplementary angles Take any integer Theory of Relative three magnitudes triangle are respectively triangle DEF triangles ABC triangles are similar vertex whole numbers
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 104 - If two similar parallelograms have a common angle, and be similarly situated ; they are about the same diameter.
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 80 - If four straight lines be proportionals, the rectangle contained by the extremes is equal to the rectangle contained by the means.
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 73 - P moves in a plane so that the ratio of its distances from two fixed points A, B in that plane is always the same.
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...