Elements of Geometry containing Books I to VI and portions of Books XI and XII of Euclid . tudes, are to one another as their bases. Let the figures be placed, so as to have their bases in theSivnie straight line ; and having drawn perpendiculars from thevertices of the triangles to the bases, the straight line, whichjoins the vertices, is parallel to that, in which their bases are,beciiuse the perpendiculars are both equal and parallel to oneanother. I. Then, if the same construction be made as in the Proposition,the demonstration will be the same. Ex. 1. ABC, DEFare two parallel straigh
Elements of Geometry containing Books I to VI and portions of Books XI and XII of Euclid . tudes, are to one another as their bases. Let the figures be placed, so as to have their bases in theSivnie straight line ; and having drawn perpendiculars from thevertices of the triangles to the bases, the straight line, whichjoins the vertices, is parallel to that, in which their bases are,beciiuse the perpendiculars are both equal and parallel to oneanother. I. Then, if the same construction be made as in the Proposition,the demonstration will be the same. Ex. 1. ABC, DEFare two parallel straight lines ; show thatthe triangle ADE is to the triangle FBG as DE is to BG. Ex. 2. If, from any point in a diagonal of a parallelogram,straight lines be drawn to the extremities of the other diagonal,the four triangles, into which the parallelogram is then divided,must be equal, two and two. 246 EUCLIDS ELEMENTS. [Book VI Proposition II. Theorem. If a straight line be drawn parallel to one of the sides of atriangle, it must cut the other sides, or those sides produced, Let DE he drawn || to BC, a side of the A must BD he to DA as CE to BE, CD. Then •/ A BDE= A CDE, on the same base DE and between the same ||s, DE, BG. I. 37. .-. A BDE is to A ADE as A CDE is to A ADE V. 6. But A BDE is to A ADE as BD is to DA, VI. 1. imd A CDE h to A ADE as CE is to EA ; BD is to DA as CE is to EA. V. 0. Ex. 1. If any two straight lines be cut by three parallellines, they are cut proportionally. (—This is of greatuse.) Ex. 2. If two sides of a quadrilateral be parallel to eachother, a straight line, drawn parallel to either of them, shallcut the other sides, or these produced, proportionally. Ex. 3. If two triangles be on equal bases, and between thesame parallels, shew that the sides of the triangles interceptequal lengths of any straight line, which is parallel to theirbases. Book VL] PROPOSITION II. 247 And Conversely, If the xidex, or the si
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