Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/362

338 Therefore, by the principles of arithmetic, the following products are equal, AF.BC.DO.AO.DB.CE and FB.CD.OA.OD.BC.EA. Therefore the following

products are equal, AF.BD.GE and FB.DC. EA. W have supposed the point to be within the triangle; if O be without the triangle two of the points D, E, F will fall on the sides produced.

60. Conversely, it may be shewn by an indirect proof that if the product AF. BD. CE be equal to the product FB.DC.EA, the three straight lines AD, BE, CF meet at the same point.

6l. We may remark that in geometrical problems the following terms sometimes occur, used in the same sense as in arithmetic; namely arithmetical progression, geometrical progression, and harmonical progression. A proposition respecting harmonical progression, which deserves notice, will now be given.

62. Let ABC he a triangle; let the angle A be bisected by a straight line which meets BC at D, and let the exterior angle at A be bisected by a straight line which meets BC produced through C, at E: then BD, BC, BE shall he in harmonical progression.