The Contents of the Fifth and Sixth Books of EuclidThe University Press, 1900 - 143 sider |
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Side xii
... Harmonic Points and Lines , of the Pole and Polar , of Inversion , of the Radical Axis and the Centres of Similitude of Two Circles , and ( so far as is possible without explaining the use of the Negative Sign in Geometry ) of Cross or ...
... Harmonic Points and Lines , of the Pole and Polar , of Inversion , of the Radical Axis and the Centres of Similitude of Two Circles , and ( so far as is possible without explaining the use of the Negative Sign in Geometry ) of Cross or ...
Side xix
... harmonic points , A and C being conjugate , and if O be the middle point of AC , then OC is a mean proportional between OB and OD . 61. If and if A : C = X : Z , B : C = Y : Z , i.e. if [ 4 , C ] = [ X , Z ] , and if [ B , C ] = [ Y , Z ] ...
... harmonic points , A and C being conjugate , and if O be the middle point of AC , then OC is a mean proportional between OB and OD . 61. If and if A : C = X : Z , B : C = Y : Z , i.e. if [ 4 , C ] = [ X , Z ] , and if [ B , C ] = [ Y , Z ] ...
Side 47
... HARMONIC POINTS . Four points A , B , C , D on a straight line are said to be four harmonic points if B and D divide AC in the same ratio , one internally and the other externally . Then A and C are called conjugate points ; as are also ...
... HARMONIC POINTS . Four points A , B , C , D on a straight line are said to be four harmonic points if B and D divide AC in the same ratio , one internally and the other externally . Then A and C are called conjugate points ; as are also ...
Side 73
... harmonic points . 47. By means of Proposition 34 construct the fourth harmonic to three given points A , B , C on a straight line ; considering separately the cases which arise according as the fourth harmonic is to be conjugate to A or ...
... harmonic points . 47. By means of Proposition 34 construct the fourth harmonic to three given points A , B , C on a straight line ; considering separately the cases which arise according as the fourth harmonic is to be conjugate to A or ...
Side 82
... harmonic points . ( ii ) Let A be the centre of a circle , B a point inside the circle , and let any chord GBF be drawn through B , and produced to H so that G , B , F , H are four harmonic points , prove that the locus of H is a ...
... harmonic points . ( ii ) Let A be the centre of a circle , B a point inside the circle , and let any chord GBF be drawn through B , and produced to H so that G , B , F , H are four harmonic points , prove that the locus of H is a ...
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The Contents of the Fifth and Sixth Books of Euclid (with a Note on ... Euclid Uten tilgangsbegrensning - 1908 |
The Contents of the Fifth and Sixth Books of Euclid Euclid,Micaiah John Muller Hill Uten tilgangsbegrensning - 1900 |
Vanlige uttrykk og setninger
arc CD BEFG BEHC bisected BLNO CÂD centre circle Commutative Law congruent corresponding sides cross-ratio DÊE draw duplicate ratio EFGH ENUNCIATION equal angles Equal Ratios equimultiples EXAMPLE expressed by Fig Fifth Book four harmonic points greater Hence the scale hypotenuse integers kind mean proportional middle point parallel to BC parallelogram point at infinity point of division PQRST Prop Proportion 66 PROPOSITION rA rB rA sB rA sC radical axis ratio compounded ratio of equality rect rectangle contained relative multiple scale required to prove respectively equal right angle second column segments side AC side corresponding similar figures similar triangles similarly described Sixth Book square ABLK supplementary angles Theory of Relative three magnitudes triangle ABC triangle are respectively triangle DEF triangles are similar unequal ratios vertex whole numbers
Populære avsnitt
Side 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.
Side 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.
Side 102 - If two similar parallelograms have a common angle, and be similarly situated ; they are about the same diameter.
Side 99 - Prove that similar triangles are to one another in the duplicate ratio of their homologous sides.
Side 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...
Side 80 - If four straight lines be proportionals, the rectangle contained by the extremes is equal to the rectangle contained by the means.
Side 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...
Side 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.
Side 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...
Side 84 - If they do not intersect, show that the radical axis is perpendicular to the line joining the centres of the circles...