Manual of plane geometry, on the heuristic plan; with numerous extra exercises, both theorems and problems, for advance work

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1891 Excerpt: ... the former. For example, HK is the projection of the line CD upon AB, and RS is the projection of RQ on NP; it is also the projection of RQ on RF. Mention all the cases of projection in Figure III., the dotted lines being perpendicular to the respective sides. 363. In any triangle the square of the side opposite an acute angle is equal to the sum of the squares of the other two sides, minus twice the product of one of those sides and the projection of the other upon that side. If C be the acute angle, then by a glance at the following diagram it will be readily seen that there may be two cases depending upon whether the projection involves an extensior of one side, which will evidently be the case if the side to be projected is opposite an obtuse angle. A A We are to prove ZB2 =BTf +ZU2-2BC. DC. In Case I. DB=BC-DC. In Case II. DB=DC-BC. The square of either of these two equations gives the same result; hence to the square add the squares of the perpendicular, and then combine the terms by using Theorem 360 (a). 364. In an obtuse triangle the square of the side opposite the obtuse angle is equal to the sum of the squares of the other two sides, plus twice the product of one of those sides and the projection of the other upon that side. Sug. Form an equation by placing the projection of the side opposite the obtuse angle equal to the sum of its two parts, then proceed as in 363. 365. If any median of a triangle be drawn, I. the sum of the squares of the other two sides is equal to twice the square of one-half the bisected side, plus twice the square of the median; and II. the difference of the squares of the other two sides is equal to twice the product of the bisected side and the projection of the median upon that side. Sug. Use 363 and 364, then combine ...