...lines may be of constant length. MISCELLANEOUS EXAMPLES ON BOOK I. 1. Prove that the angle between the bisectors of two consecutive angles of a quadrilateral is equal to half the sum of its other two angles. 2. Prove that the sum of the distances of any point taken within...

...-7-5 — — j from the centre. 12. Every body has one and only one centre of gravity. Hence show that the lines joining the middle points of the opposite sides of a quadrilateral bisect each other. (Consider four equal particles at the vertices). 13. From the vertices of a given triangle let perpendiculars...

...Two quadrilaterals whose diagonals are respectively equal and form equal angles are equivalent. 157. Lines joining the middle points of the opposite sides of a quadrilateral bisect each other. 158. If lines are drawn from the vertices of a square bisecting the opposite sides in order, there...

...distance «t from the centre. -R'-r2 1 2. Every body has one and only one centre of mass. Hence show that the lines joining the middle points of the opposite sides of a quadrilateral bisect each other. (Consider four equal particles at the vertices.) 13. From the vertices of a given triangle let perpendiculars...

...Two quadrilaterals whose diagonals are respectively equal and form equal angles are equivalent. 157. Lines joining the middle points of the opposite sides of a quadrilateral bisect each other. 158. If lines are drawn from the vertices of a square bisecting the opposite sides in order, there...

...parallelogram, to prove the parallelogram one-fourth of the quadrilateral, using (24). 54. The lines which join the middle points of the opposite sides of a quadrilateral, bisect each other. (Corollary of 53. ) 55. If, through the extremities of each diagonal of a quadrilateral, parallels...

...parallelogram ; and its area is half that of the quadrilateral. [Consider Prop. 1.] 3. The straight lines joining the middle points of the opposite sides of a quadrilateral bisect each other. [This follows immediately from the last, if it can be shewn that the diagonals of a parallelogram bisect...

...distance cr3 -= ; from the centre. 12. Every body has one and only one centre of mass. Hence show that the lines joining the middle points of the opposite sides of a quadrilateral bisect each other. (Consider four equal particles at the vertices.) 13. From the vertices of a given triangle let perpendiculars...

...middle points of the opposite sides, and thus show that the three points are coincident. 10. Prove that the lines joining the middle points of the opposite sides of a quadrilateral and the line joining the middle points of the diagonals all bisect each other. To do this, express...

...H are the middle points of AD, CD, BC, and AB respectively, prove that EFGH is a parallelogram. 8. The lines joining the middle points of the opposite sides of a quadrilateral bisect the line joining the middle points of the diagonals. 9. If D and E are the middle points of the sides...