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ABCD Analysis.—Let angular points anharmonic ratio bisected bisectors centre of mean centre of similitude chord circle touching circumference circumscribed common tangent concurrent concurrent lines conjugate points Cor.—If cut orthogonally Dem.—Let Dem.—The denoted diameter divided draw equal equiangular equianharmonically Euclid external fixed point four points Geometry given circle given in position given in species given line given point Given the base harmonic conjugates harmonic mean harmonic system Hence the Proposition homographic inverse involution let fall limiting points line joining locus mean position meet middle point Nine-points Circle opposite sides orthogonally pairs passes pencil perpendicular point of intersection points of contact polar pole polygon Prop quadrilateral radical axis radii radius rays rectangle contained regular polygon respect right line right Zs Section segments semicircle square system of points theorem three given transversal triangle vertical angle
23 ページ - Three times the sum of the squares of the sides of a triangle is equal to four times the sum of the squares of the medians.
19 ページ - ABC a perpendicular is drawn to the opposite side, meeting it, produced if necessary, at D ; from the angle B a perpendicular is drawn to the opposite side, meeting it, produced if necessary, at E: shew that the straight lines which join...
28 ページ - A, and the adjacent angular points of the squares joined, the sum of the squares of the three joining lines is equal to three times the sum of the squares of the sides of the triangle.
27 ページ - Find the locus of a point the sum of the squares of whose distances from two given points is constant.
149 ページ - JL from the right angle on the hypotenuse of a rightangled A is a harmonic mean between the segments of the hypotenuse made by the point of contact of the inscribed circle. 10. If a line be cut harmonically by two Os, the locus of the foot of the _L, let fall on it from either centre, is a O, and it cuts any two positions of itself homographically (see Prop. 3, Cor. 2, Section VII.)11.
58 ページ - If the' great circle and the circumscribed polygon revolve together about AA', the great circle will describe the surface of a sphere, the angular points of the polygon except A, A' will move round the surface of a larger sphere, the points of contact of the sides of the polygon with the great circle of the inner sphere will describe circles on that sphere in planes perpendicular to AA', and the sides of the polygon themselves will describe portions of conical surfaces. The circumscribed figure will...
110 ページ - O'P + O'Q = PQ = r ; E + 8 1 ETS" E-8 1 a result already proved by a different method (see Prop. 11, Section I.). Prop. 13.—If a variable chord of a circle subtend a right angle at a fixed point, the locus of its pole is a circle. chord •which subtends a right Z at a fixed point P; AE, BE tangents at A and B, then E is the pole of AB : it is required to find the locus of E. Let O be the centre of X. Join OE, intersecting AB in I; then, denoting the radius of X by r, we have OP...
45 ページ - Two points, such as A and B, which possess the property that the polar of either passes through the other, are called conjugate points with respect to the circle, and their polars are called conjugate lines. Prop.