GEOMETRICAL PROBLEMS, 
34? 
PROBLEM XXXVI. 
To male a right-angled triangle equal to a given square 
ABCD, ivhose sides shall be in arithmetical progression, 
CONSTRUCTION. 
3 AB 
In A3 produced take BF r ■■ ■ o —, and upon AF 
describe the semi-circle AEF, cutting BC produced 
in E ; 
done. 
DEMONSTRATION, 
Since, by construction, QB : BE ; : 4 r 3, therefore 
will Bd : EB 2 : : 16 : 9, and BQ* + BE 2 : BE 2 : : 16 
4- 9 (25) : 9, that is, EQ a : BE 1 :: 25 : 9 I Eue. 47- 1.); 
whence EQ : BE.: 5 : 3 [Eue. 22. 6.); therefore the 
sides BE, BQ, and EQ, being to one another in the 
1’atio of the numbers 3, 4, and 5, are in arithmetical 
4EB 
progression. And, because BQ is = ——, thence will 
EB / BQ _ 2EB 1 
2 3 
2BF x AB 
3 
= AB 
Q. E. U, 
METHOD OF CALCULATION. 
Seeing BF is =•: (BEi/ABaBF) will be x: 
Z 
ABv/T; whence BQ and EQ ) will 
be likewise given,