SB Mr, glen ie’s Propofitions on the 
cor. i. The alternate triangles of thofe four, which 
have their vertices in the point, where the right line 
drawn from the right angle meets the hvpotenule, are 
fimilar, and have to each other the proportion of the 
fegments of the hypotenule, or the duplicate proportion 
of the Tides containing the right angle. 
cor. ii. Either pair of the adjacent triangles lying on 
different Tides of the right line drawn from the right 
angle, and having their vertices in the interfedtion of the 
right lines drawn from the angles at the hypotenufe, 
have to each other the proportion of the alternate trian- 
gles, having their vertices in the interfedtion of the lirit- 
mentioned line and the hypotenufe. 
cor. hi. The trapezium or quadrilateral figure formed 
by the fegments of the fides adjacent to the right angle, 
and the right lines joining their extremities with the in- 
terfedtion of the hypotenufe, and the right line drawn 
from the right angle to meet it, is capable of being in- 
fcribed in a circle ; and is divided at the interfedtion of 
right lines drawn from the angles at the hypotenufe to 
the alternate angles of fquares, defcribed on the fides 
containing the right angle, into triangles which are pro- 
portional to one another, and when taken two by two, as 
Further, the angles, pla, clq^, are equal, each being half a right angle. And 
the angles pal, lcq^, are equal (by 8. Elem.6.). Therefore the angles apl, cql, 
will be equal, and the triangles apl, cql, fimilar, and the Tides fubtending the 
equal angles proportional. Therefore, pl : la — ql : lc. By permutation, 
lp : lq^= la : lc. But la : lc — ac : cb (by 8. Elem. 6.). Therefore, 
j.p : lq^ ac : cb. 4L E. D. s. horsley. 
they 
