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This equation emerged while I was solving a geometry problem, which was equivalent to proving $x+n=m$.

Solve the system of polynominal equations where $x,y,z,m,n,l$ are positive reals.

$$ \begin{cases} yz=nl\\ x^2=m(m+n)\\ (x+z)^2=m(m+n+l)\\ (x+z)^2+z^2=(m+n)^2\\ (x+z)^2+(y+z)^2=(x+y)^2 \end{cases} $$

Through complex calculation, I solved the equation for $z$, which gave me $x=\frac{\sqrt {7}+1}{3}z$, $m=\frac{\sqrt {7}+1}{2}z$, $n=\frac{\sqrt {7}+1}{6}z$.

However, this way seemed a bit complex, and solving it for other variables did not seem simpler either.

Is there any simple way to prove $x+n=m$ without calculating each value?

An Link to the Original Geometry Problem

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Chad Shin
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  • What was exactly the original problem? – sinbadh Jan 18 '16 at 03:59
  • Angle $C$ is a right angle in triangle $ABC$, and the incircle is tangent to lines $AC,CB,BA$ in $E,D,F$. $AD$ meets the incircle in point $P$, and the angle of $BPC$ is right. Prove that $AE+AP=PD$. While I think that there is a geometric solution to this, I am also interested in a easier algebraic solution. (I am not very famaliar with the english words for geometry, so excuse any errors that I have made.) – Chad Shin Jan 18 '16 at 04:04
  • You should seek a geometric answer to the problem, as the problem itself is geometric. I understand your desire to find an algebraic solution, but since this is a geometry problem, you may not find such a solution. – K. Jiang Jan 18 '16 at 05:17
  • @K.Jiang Is it all right if I asked the geometry question in a seperate question? I am just curious about the algebraic way. – Chad Shin Jan 18 '16 at 08:42
  • I don't see why not. I just want you to receive the easiest answer possible (algebra can get very messy). – K. Jiang Jan 18 '16 at 13:59
  • (1) $x+n=m\iff x-m=-n$; (2) $x^2=m(m+n)\iff (x-m)(x+m)=mn$ Hence, from (1) and (2), $x=-2m$ – Piquito Jan 21 '16 at 13:00
  • @Piquito Note that $x+n=m$ is not given. We have to prove it. Also, your equation has the solution $3x=2m$ as well. – Chad Shin Jan 21 '16 at 13:03
  • @ChadShin: What I did was a remark: if $x+n=m$ true then .... I shall pay attention for a while to your problem. Regards. – Piquito Jan 21 '16 at 15:00
  • @Piquito Your attention is appreciated. – Chad Shin Jan 21 '16 at 16:38

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