The Contents of the Fifth and Sixth Books of EuclidThe University Press, 1900 - 143 σελίδες |
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Σελίδα 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 ...
Σελίδα 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 A : C = X : Z , i.e. if [ A , C ] = [ X , Z ] , and if B : C = Y : 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 A : C = X : Z , i.e. if [ A , C ] = [ X , Z ] , and if B : C = Y : Z , and if [ B , C ] = [ Y , Z ] ...
Σελίδα 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 ...
Σελίδα 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 ...
Σελίδα 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 ...
Άλλες εκδόσεις - Προβολή όλων
The Contents of the Fifth and Sixth Books of Euclid Euclid,Micaiah John Muller Hill Πλήρης προβολή - 1900 |
Συχνά εμφανιζόμενοι όροι και φράσεις
ABCD ABEF angle equal angles reciprocally proportional BCGE BCHD BEFG BEHC bisected BLNO centre circle corresponding sides cross-ratio cutting AC DÊE definition divided drawn perpendicular duplicate ratio ENUNCIATION EQPS equal angles equimultiples Euclid's EXAMPLE exhibited in Fig expressed by Fig fact exhibited four harmonic points four straight lines greater Hence by Prop Hence the triangles hypotenuse integer inversion kind locus mean proportional middle point parallel to BC parallelogram PQRST PROPOSITION Proposition 48 rA rB rA sC radical axis ratio compounded ratio of equality rect rectangle contained relative multiple scale required to prove respectively equal right angle segments shows the fact side BC side corresponding similar figures similar triangles similarly described square on AB square on AC supplementary angles tangents three magnitudes triangle ABC triangle DEF triangles are similar vertex
Δημοφιλή αποσπάσματα
Σελίδα 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.
Σελίδα 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.
Σελίδα 99 - Prove that similar triangles are to one another in the duplicate ratio of their homologous sides.
Σελίδα 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...
Σελίδα 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...
Σελίδα xvi - In a right triangle, if a perpendicular is drawn from the vertex of the right angle to the hypotenuse : I.
Σελίδα 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...
Σελίδα 84 - If they do not intersect, show that the radical axis is perpendicular to the line joining the centres of the circles...
Σελίδα 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.
Σελίδα 80 - If four straight lines be proportionals, the rectangle contained by the extremes is equal to the rectangle contained by the means.