...the sum of each body into the square of its distance from this point, is a constant quantity. 12. The sum of the squares of the distances of the centre of gravity of any number of equal bodies, from the centre of gravity of each, is equal to the sum of the squares...

...of the squares be joined ; the sum of the squares of the sides of the hexagonal figure thus formed is equal to four times the sum of the squares of the sides of the triangle. 14,. If all the angular points of a regular polygon of n sides be joined, and...

...+ BC* + BD* + CD1 - *(AG* + BGF + CG2 + Hence the sum of the squares of the six edges of a pyramid is equal to four times the sum of the squares of the distances of its angular points from its centre of gravity. 168. When a system of bodies is in equilibrium under...

...of the squares be joined ; the sum of the squares of the sides of the hexagonal figure thus formed is equal to four times the sum of the squares of the sides of the triangle. GEOMETRICAL EXERCISES ON BOOK III. THEOREM I. If AB, CD be chords of a circle...

...proposition it may be proved, that the sum of the squares of the four diagonals of the parallelepiped BN is equal to four times the sum of the squares of the three edges about one of the solid angles ; that is, We have already, as above, the value of OP2. Now...

...will pass through a given point. 86. By Amicus. In every tetrahedron, the sum of the squares of the six edges is equal to four times the sum of the squares of the lines which join the middles of the opposite edges. [Our readers are requested to furnish any other...

...the square on half the base. 8. Three times the sum of the squares on the three sides of a triangle is equal to four times the sum of the squares of the Hues drawn from the angles to bisect the opposite sides. 9. The difference between the squares on two...

...of the squares be joined : the sum of the squares of the sides of the hexagonal figure thus formed, is equal to four times the sum of the squares of the sides of the triangle. 23. If all the angular points of a regular polygon be joined, and r be the radius...

...triangle, and the angular points be joined, the sum of the squares of the hexagonal figure thus formed, is equal to four times the sum of the squares of the sides of the triangle. 5. If from a point P, in the diameter AB of a circle, PQ, PR be drawn to the...

...be inscribed in the two solids. 10. The sum of the squares of the four diagonals of a parallelopiped is equal to four times the sum of the squares of the edges. 11. If with each angular point of any parallelopiped as centres equal spheres be described,...