rOL. LXVI.] PHILOSOPHICAL TRANSACTIONS. . 733 



be drawn to the points where these lines meet the sides; the lines so drawn will 

 make equal angles with the hypothenuse, and the right line drawn from the right 

 angle to meet it; and will also have to each other the proportion of the sides con- 

 taining the right angle. 



Cor. 1. The alternate triangles of those 4, which have their vertices in the 

 point where the right line drawn from the right angle meets the hypothenuse, are 

 similar, and have to each other the proportion of the segments of the hypothe- 

 nuse, or the duplicate proportion of the sides containing the right angle. 



Cor. 1. Either pair of the adjacent triangles lying on different sides of the 

 right line drawn from the right angle, and having their vertices in the intersec- 

 tion of the right lines drawn from the angles at the hypothenuse, have to each 

 other the proportion of the alternate triangles, having their vertices in the inter- 

 section of the first-mentioned line and the hypothenuse. 



Cor. 3. The trapezium or quadrilateral figure formed by the segments of the 

 sides adjacent to the right angle, and the right lines joining their extremities with 

 the intersection of the hypothenuse and the right line drawn from the right angle 

 to meet it, is capable of being inscribed in a circle; and is divided at the inter- 

 section of right lines drawn from the angles at the hypothenuse to the alternate 

 angles of squares, described on the sides containing the right angle, into tri- 

 angles which are proportional to each other, and when taken two by two, as 

 they lie adjacent on different sides of the diagonal, are proportional to the un- 

 equal sides of the trapezium, and to the two triangles into which the diagonal 

 divides it. 



Prop. Q. If from the angles at the base of any right lined triangle, right lines 

 be drawn to the alternate angles of rhomboids described on the other two sides, 

 and reciprocally applied to them produced; a right line drawn from the vertex, 

 through the intersection of these lines, will cut the base into two parts, having to 

 each other the proportion compounded of the proportion of the sides, and of the 

 proportion of the other two lines comprehending the rhomboids. — Let the triangle 

 be ACB, the base AB, the rhomboids acep, bcdg, fig. 11, pi. 13; and let the 

 right lines bf, ag, be drawn. Then, if from the vertex c through their inter- 

 section o, a right line col be drawn to meet the base, the segments al, lb, will 

 have to each other the proportion compounded of the proportions of ac to CB, 

 and of CE to cd. 



Scholium. If ce, cd, be equal to each other, then al has to lb the propor- 

 tion of AC to CB, and cl bisects the angle acb; if ce have to cd the inverse 

 proportion of ac to cb, al is equal to lb; if ce have to cd the proportion of 

 AC to CB, AL has to LB the duplicate proportion of ac to cb; and universally, if 

 CE have to cd any multiplicate proportion, n, of ac to CB, al has to lb such a 

 multiplicate proportion of ac to cb as is expressed by the number w -f- !• And 



