greetings and a question!

*To*: mathgroup at smc.vnet.net*Subject*: [mg83116] greetings and a question!*From*: dimitris <dimmechan at yahoo.com>*Date*: Sun, 11 Nov 2007 03:02:51 -0500 (EST)

Hello to all of you! Unfortunately, family, working and research issues prevent me from participating to the forum as frequent as I used to. Anyway... It is my first post since a long time so everybody be patient! A student of mine came across the following equation in a mathematical contest: In[1]:= eq = x^2 + y^2 + (a + b)*x - (a - b)*y + a^2 + b^2 - a - b + 1==0; (all variables are assumed real) Of course for Mathematica the solution is rather trivial. In[1]:= $Version Out[1]= "5.2 for Microsoft Windows (June 20, 2005)" In[2]:= eq = x^2 + y^2 + (a + b)*x - (a - b)*y + a^2 + b^2 - a - b + 1==0; In[3]:= Reduce[eq, {a, b, x, y}, Reals] ToRules[%] eq /. % Out[3]= a == 1 && b == 1 && x == -1 && y == 0 Out[4]= {a -> 1, b -> 1, x -> -1, y -> 0} Out[5]= True Can somebody explain concisely the mathematica concept behind this solution? In fact I would be much obliged if somebody pointed me out how to obtain the result by hand. Also, by curiosity, how Mathematica reaches the result? Dimitris

**Follow-Ups**:**Re: greetings and a question!***From:*Andrzej Kozlowski <akoz@mimuw.edu.pl>