AA\ 



\»-A\^A<^VMv>/ro7?i the Definitions of Euclid, 



373 



until the produced parts are equal to AB and AC. Draw DB, 



DE, and EC. The angles BAC and BAD 



are equal to two right angles (XIII.)*, and 



are therefore equal to each other ; the angles 



DAE and CAE are also equal to two right 



angles; therefore the four angles at A are 



right angles. In the triangles BAC and 



BAD, two sides of the one (BA and AC) are 



equal to two sides of the other (BA and AD), 



and the included angles are equal ; hence BC is equal to BD, and 



the angles ABC and ACB are equal to ABD and ADB. In the 



same way the sides DE and EC can be shown to be equal to 



each other and to DB and BC ; and the angles in each triangle 



to be respectively equal. Hence (from the definition of a square) 



these four angles are right angles ; and ABC and ACB are half 



right angles. Therefore the right-angled isosceles triangle ABC 



contains two right angles. 



Lemma I. If one of the angles (ABC) at the hypothenuse of a 

 I ^^ right-angled, triangle be equal to half a right angle j the tri- 

 ,^ angle is isosceles. 



Y\ If AB and AC are not equal, let AC be 

 inade equal to AB ; and draw BD. 



As BAD is an isosceles right-angled 

 triangle, the angle ABD is equal to half a 

 right angle. But ABC is equal to half a 

 right angle ; therefore a part is equal to 

 the whole, which is absurd; therefore AC must be equal to AB. 



Lemma II. If in two triangles (ABC and DEE), two sides of 

 the one (AB and BC) be respectively equal to two sides of the 

 other (DE an^EF), and an angle (BAC), greater than a right 

 angle, in the one be equal to an angle (EDF) in the other, the 

 two triangles are equal in every respect. 



Let the triangles be applied to each other so that the bases 

 \ 1^ .bDY*jh|«Kr> s'lari 



BC and EF coincide. Then DE will lie on AB. If not, let 

 DEF take the position BDC. Join A and D. 



As AB and BD are equal, the angles DAB and ADB are 



* The references are made to Mr. Potts' large edition of Euclid. 



