130 PROCEEDINGS OF THE AMERICAN ACADEMY 



to E, so that DE = AD, and that the point E and the angle C are 

 connected hy a right line ; then shall the sum of the angles of the tri- 

 angle ACE equal that of the triangle ABC. 



" For the opposite vertical angles A D B, CBE are equal (Sim- 

 son's Euclid, B. I., prop. 15), and by construction BD = D C, AD 

 =^DE; hence (Sim., B. I., p. 4) the triangles ADB, CDE are 

 identical ; so that their bases A B, C E are equal, and their angles 

 A B D, E C B are equal ; also the angle BAD equals the angle 

 CED. Hence the angle C of the triangle ACE equals the sum 

 of the two angles B and C of the given triangle (il 5 C), and the 

 sum of the angles A and E of the triangle ^ C £ is equal to the 

 angle A of the given triangle {ABC)-, .'. the sum of the angles of 

 the triangle A C E Is equal to that of the given triangle ABC, as 

 required. 



" Cor. 1. Let the angle jB ^ C of the given triangle be denoted by 

 A ; then '\( C E {= A B) is not greater than A C, the angle C ^ £ is 

 not greater than C EA (Sim., props. 5, 19, B. I.) ; hence the angle 

 A of the triangle A C E is not greater than ^ ; we shall call the tri- 

 angle ACE the first derived triangle. Of the two sides A C and 

 C £ of the triangle ^ C £, let C E be that which is not the greater ; 

 and let a right line be drawn from the angle A, opposite to the side 

 C £, through the point, H, of bisection of C £, and suppose the line 

 thus drawn to be produced in the direction A H to I, so that H 1= 

 HA ; then connect the point /and the angle C of the triangle ACE 

 by a right line ; and there will be formed the triangle ACL In the 

 same way that it was shown that the sum of the angles of the triangle 

 ^ C £ is equal to the sum of the angles of the (given) triangle ABC, 

 it may be shown that the sum of the angles of the triangle ^ C 7 is 

 equal to that of ACE; consequently the sum of the angles of ^ C J 

 equals that of the given triangle ABC. And if ^ C is not greater 

 than C 7, then it may be shown (as before) that the angle I is not 

 greater than the angle CAE^2, and since CA E is not greater than 

 f, .-. the angle I is not greater than |;, we shall call A CI the second 

 derived triangle. 



" We may in the same way (that we derived the triangle A CI from 

 ACE) derive a triangle from A CI (called the third derived trian- 

 gle), having the sum of its angles equal to that of ^ C I, and of course 

 equal to that of the given triangle ABC, and having one of its angles 

 not greater than the angle ^ I C -i- 2, and consequently not greater 



