Unable to find solution to 6 simultaneous equations

I have 6 equations: $$\begin{array}{l} x_1^2+y_1^2=(x_2-x_1)^2+(y_2-y_1)^2 \\ x_3^2+(3-y_3)^2=(x_3-x_1)^2+(y_3-y_1)^2 \\ 4[(x_3-x_2)^2+(y_3-y_2)^2]=(4-x_2)^2+y_2^2 \\ x_2^2+y_2^2=4(x_1^2+y_1^2)\\ 4[x_3^2+(3-y_3)^2]=x_1^2+(3-y_1)^2\\ 9[(x_3-y_2)^2+(y_3-y_2)^2]=(4-x_3)^2+y_3^2 \end{array}$$

that I know (if equations are set up correctly) has a real solution inside the right triangle {{0,0},{0,3},{4,0}}. However, NSolve returns the null solution. I was wondering if someone could look at my code and confirm if I am coding it correctly or if NSolve perhaps cannot solve it, suggest an alternate method I could use to do so?

Thanks.

r1 = Polygon[{{0, 0}, {0, 3}, {4, 0}, {0, 0}}];
eqn1 = x1^2 + y1^2 == (x2 - x1)^2 + (y2 - y1)^2;
eqn2 = x3^2 + (3 - y3)^2 == (x3 - x1)^2 + (y3 - y1)^2;
eqn3 = 4 ((x3 - x2)^2 + (y3 - y2)^2) == (4 - x2)^2 + y2^2;
eqn4 = x2^2 + y2^2 == 4 (x1^2 + y1^2);
eqn5 = 4 (x3^2 + (3 - y3)^2) == x1^2 + (3 - y1)^2;
eqn6 = 9 ((x3 - x2)^2 + (y3 - y2)^2) == (4 - x3)^2 + y3^2;
NSolve[{eqn1 && eqn2 && eqn3 && eqn4 && eqn5 && eqn6 &&
Element[ {x1, x2, x3, y1, y2, y3}, r1]}, {x1, x2, x3, y1, y2, y3}]

There are a few issues with your code.

1. Region constrains are incorrect.
2. Should use Solve instead of NSolve (numerical approximation).

The region constrain is in $$\mathbb{R}^2$$ but your syntax is for an $$\mathbb{R}$$ constraint. To check the pairs are in $$\mathbb{R}^2$$.

Element[Alternatives @@ {{x1, y1}, {x2, y2}, {x3, y3}}, r1] Use this constraint with Solve produces the answer.

Solve[{eqn1 && eqn2 && eqn3 && eqn4 && eqn5 && eqn6 &&
Element[Alternatives @@ {{x1, y1}, {x2, y2}, {x3, y3}}, r1]},
{x1, x2, x3, y1, y2, y3}]
{{x1 -> 4/5, x2 -> 8/5, x3 -> 2/5, y1 -> 3/5, y2 -> 6/5, y3 -> 9/5}}

Hope this helps.

PS: Note that you only need to add the ordered vertices for Polygon as the closure is applied between the first and last vertices. So r1 can be defined as Polygon[{{0, 0}, {0, 3}, {4, 0}}].

• You guys are tops in here! :) Sep 14 '19 at 13:32
• I should point out it took about 40 min for Solve to find the solution above on my machine at 2.2 GHz running ver. 11.1. Sep 14 '19 at 19:47
• @Dominic Its about 2 seconds in 12.0 on my 4 yr old laptop (2.4 GHz). Sep 14 '19 at 20:06