How to solve this eliptic equation?

Question

I want to find a solution to

$$ \frac{x_1}{x_2 + x_3} +\frac{x_2}{x_1 + x_3} + \frac{x_3}{x_1+x_2} = 4 $$

for $x_1,x_2,x_3>0$, and $x_1+x_2+x_3=1$.

We have $$\frac{x_1}{1-x_1}+\frac{x_2}{1-x_2}+\frac{1-x_1-x_2}{x_1+x_2}=4.$$

Let $\dfrac{x_1}{1-x_1}=s,\dfrac{x_2}{1-x_2}=t,$then $x_1=\dfrac{s}{1+s},x_2=\dfrac{t}{1+t}.$ We then have $$s+t+\dfrac{1-st}{s+t+2st}=4,$$ and $s=x+y,t=x-y,$then$$\frac{4 x^3+3 x^2-4 x y^2+y^2+1}{2 \left(x^2+x-y^2\right)}=4$$ $$y^2=\frac{4 x^3-5 x^2-8 x+1}{4 x-9}$$ which eventually simplifies to solving $$y^2 = x^3 + 121 x^2 + 1144 x + 2704.$$ How can we solve this? I am unfamiliar with eliptic curves, even after reading some basic material, it seems each equation is tackled differently. Is there a routine method for equations like this?

Answer 1

First notice that x=0,y=52 is a solution. Take a tangent to the curve at this point and it will necessarily intersect at another rational point. By successively taking intersections of the curve with either tangents of rational points or the lines passing through two different rational points you can find more rational points.

Answer 2

Let $x_3 = 1-x_1-x_2.$ We get $$(6-7x_1)x_2^2+(-7x_1^2+13x_1-6)x_2+6x_1^2+1-6x_1=0$$

Since $x_2$ must be rational number then discriminant must be square number.

Hence we get

$$V^2 = 49U^4-14U^3-59U^2+16U+12\tag{1}$$ with $$U=x_1$$ An equation $(1)$ is birationally equivalent to the elliptic curve
$$Y^2 + XY + Y = X^3 - 234X + 1352 \tag{2}$$ with $$U = \frac{-X-3}{X-23}, V = \frac{-52Y-26X-26}{(X-23)^2}$$ According to LMFDB , elliptic curve has rank $1$ with generator $(8,-1).$

Let $P(X,Y)=(8,-1)$ then $13P$ gives positive solution using group law as follows.

$x_1 = \frac{360580518798960218928732302909807349160733186141387092674568680456952454542061941533770232294581808506248091531}{5049549479055895370585527032143752370420166865650725292672977625843429614339787186642533620730887839277641883695}$

$x_2 = \frac{306595938811259225832173528708283063181310906692288470156330531143167380161534287576910178181384187701055520003}{2212455878539701410315253838580846315686580343082895422240759011653186431824457611838047222821239666672118415365}$

$x_3 = \frac{3852086946706166284829913202913539977730505088064544668668081343553155180751922744069311820918853233969099047039}{4875970872038496234103522347007495376171052920878501712857733921453635116079760054850598705847152516034789194083}$