234^ ON TRIANGLES. 



hairs that grow on the plant of the wheat whilft young, but 

 which fall oft" when it ripens, magnified to the fame degree 

 as the preceding figures. 



II. 



Theorems refpe&ing the Properties of the Sides of Triangles in- 

 terfecled by Bight Lines drazvn from the three Angles jo as to 

 meet in one Point. By Mr. John Go ugh. 



To Mr. NICHOLSON. 

 SIR, 



MiddleJIiais, March 1, 1S05. 



Certain proper- V^rEOMETRICI ANS have paid but little attention to the 

 ties of triangles, properties of the fides of triangles, which are interfered by 

 right lines, drawn from the three oppofite angles foas to meet 

 in one point. Perhaps all the theorems of this kind that have 

 been hitherto publimed, are contained in (he fecond book of 

 Emerfon's Geometry, with the (ingle exception of a propor- 

 tion given by Mr. Landen; who has proved that if one of the 

 internal angles of a triangle, and the two oppofite external an- 

 gles of the fame be bife&ed, the right lines, drawn for this 

 purpofe, will alfo meet in a common point. The following 

 propofilions take a more general view of' the fubject ; and as 

 fome mathematical papers have appeared of late in the Philo- 

 fophical Journal, I have ventured to afka place for the prefent 

 Aieet, in your valuable mifcellany. 



I remain, &c. 



JOHN GOUGH. 



P. S. In vol. X. page 66, line 2, for j| yards read |3§ yards. 



Propofition 1JI. Let three right lines CM, B N and A S, 

 fP/a/eXT. Fig. 1, 2.) drawn from the three angles of a trian- 

 gle ABC, interfeel the oppofite fides, produced or not, in 

 the points M, N and S, and alfo meet each other in a com- 

 mon point O; then if all the points M, N and S lay betwixt 



the 



