Assuming seems not working and "Solve::svars: Equations may not give solutions for all "solve" variables." issue

Question

I am trying to solve these equations

    Assuming[{x1 ∈ Reals && x1 > 0 && x1 < 1}, 
 Solve[{(a3 - r2)^2 + (b3 + r2 - 1)^2 == 
    r2^2, (a6 - r2)^2 + (b6 + r2 - 1)^2 == r2^2, 
   b5 == k1*(a5 - 1) + 1, b3 == k1*(a3 - 1) + 1, 
   b2 == k1*(a2 - 1) + 1, b6 == k2*(a6 - a1) + b1, 
   b4 == k2*(a4 - a1) + b1, (a2 - 1/2)^2 + (b2)^2 == 
    1/4, (a1 - 1/2)^2 + (b1)^2 == 1/4, (a4 - x1)^2 + (b4 - y1)^2 == 
    r1^2, x1 + r1 == 1, (a5 - x1)^2 + (b5 - y1)^2 == 
    r1^2, (y1 - b4)*k2 == -(x1 - a4), (b1)*k2 == -(a1 - 1/2), (b2)*
     k1 == -(a2 - 1/2), (1 - r2 - b3)*k1 == -(r2 - a3), (1 - r2 - b6)*
     k2 == -(r2 - a6), (b5 - y1)*k1 == -(a5 - x1)}, {x1, y1, r1, r2, 
   k1, k2, a1, a2, a3, a4, a5, a6, b1, b2, b3, b4, b5, b6}]]

And it does output a valid solution

x1 -> 5/6, y1 -> 2/3, r1 -> 1/6, r2 -> 1/4, k1 -> 3/4, k2 -> -(4/3),
a1 -> 9/10, a2 -> 1/5, a3 -> 2/5, a4 -> 7/10, a5 -> 11/15, a6 -> 9/20, 
b1 -> 3/10, b2 -> 2/5, b3 -> 11/20, b4 -> 17/30, b5 -> 4/5, b6 -> 9/10

But it also outputs a bunch of other invalid/undesired "solutions". I was trying to limit the output by using Assuming, apparently it does not work though.

Also, the console outputs

Solve::svars: Equations may not give solutions for all "solve" variables.

However, I clearly gave 18 variables with 18 equations. Why so?

My mma is 9.0.0.0. on mac.

Answer 1

Using v 9.0 on a Mac

$Version

"9.0 for Mac OS X x86 (64-bit) (November 20, 2012)"

eqns = {(a3 - r2)^2 + (b3 + r2 - 1)^2 == r2^2, (a6 - r2)^2 + (b6 + r2 - 1)^2 == r2^2, b5 == k1*(a5 - 1) + 1, b3 == k1*(a3 - 1) + 1, b2 == k1*(a2 - 1) + 1, b6 == k2*(a6 - a1) + b1, b4 == k2*(a4 - a1) + b1, (a2 - 1/2)^2 + (b2)^2 == 1/4, (a1 - 1/2)^2 + (b1)^2 == 1/4, (a4 - x1)^2 + (b4 - y1)^2 == r1^2, x1 + r1 == 1, (a5 - x1)^2 + (b5 - y1)^2 == r1^2, (y1 - b4)*k2 == -(x1 - a4), (b1)*k2 == -(a1 - 1/2), (b2)*k1 == -(a2 - 1/2), (1 - r2 - b3)*k1 == -(r2 - a3), (1 - r2 - b6)*k2 == -(r2 - a6), (b5 - y1)*k1 == -(a5 - x1)};

sol = Solve[eqns, {x1, y1, r1, r2, k1, k2, a1, a2, a3, a4, a5, a6, b1, b2, b3, b4, b5, b6}];

Solve::svars: Equations may not give solutions for all "solve" variables. >>

Length[sol]

20

Assuming that you are looking for real solutions:

solr = Select[sol, FreeQ[N[#], Complex[_, _]] &];

Length[solr]

12

Some solutions are dependent on y1. Removing those,

solr2 = Select[solr, Length[#] == 18 &]

{{x1 -> 1/2, y1 -> 0, r1 -> 1/2, r2 -> 1/4, k1 -> 3/4, k2 -> -(4/3), a1 -> 9/10, a2 -> 1/5, a3 -> 2/5, a4 -> 9/10, a5 -> 1/5, a6 -> 9/20, b1 -> 3/10, b2 -> 2/5, b3 -> 11/20, b4 -> 3/10, b5 -> 2/5, b6 -> 9/10}, {x1 -> 2, y1 -> 1/2, r1 -> -1, r2 -> 1/4, k1 -> 3/4, k2 -> -(4/3), a1 -> 9/10, a2 -> 1/5, a3 -> 2/5, a4 -> 6/5, a5 -> 7/5, a6 -> 9/20, b1 -> 3/10, b2 -> 2/5, b3 -> 11/20, b4 -> -(1/10), b5 -> 13/10, b6 -> 9/10}, {x1 -> 5/6, y1 -> 2/3, r1 -> 1/6, r2 -> 1/4, k1 -> 3/4, k2 -> -(4/3), a1 -> 9/10, a2 -> 1/5, a3 -> 2/5, a4 -> 7/10, a5 -> 11/15, a6 -> 9/20, b1 -> 3/10, b2 -> 2/5, b3 -> 11/20, b4 -> 17/30, b5 -> 4/5, b6 -> 9/10}, {x1 -> 2/3, y1 -> 7/6, r1 -> 1/3, r2 -> 1/4, k1 -> 3/4, k2 -> -(4/3), a1 -> 9/10, a2 -> 1/5, a3 -> 2/5, a4 -> 2/5, a5 -> 13/15, a6 -> 9/20, b1 -> 3/10, b2 -> 2/5, b3 -> 11/20, b4 -> 29/30, b5 -> 9/10, b6 -> 9/10}, {x1 -> 1/2, y1 -> 0, r1 -> 1/2, r2 -> 1/4, k1 -> 3/4, k2 -> 0, a1 -> 1/2, a2 -> 1/5, a3 -> 2/5, a4 -> 1/2, a5 -> 1/5, a6 -> 1/4, b1 -> 1/2, b2 -> 2/5, b3 -> 11/20, b4 -> 1/2, b5 -> 2/5, b6 -> 1/2}, {x1 -> 5/6, y1 -> 2/3, r1 -> 1/6, r2 -> 1/4, k1 -> 3/4, k2 -> 0, a1 -> 1/2, a2 -> 1/5, a3 -> 2/5, a4 -> 5/6, a5 -> 11/15, a6 -> 1/4, b1 -> 1/2, b2 -> 2/5, b3 -> 11/20, b4 -> 1/2, b5 -> 4/5, b6 -> 1/2}, {x1 -> 4/3, y1 -> 5/6, r1 -> -(1/3), r2 -> 1/4, k1 -> 3/4, k2 -> 0, a1 -> 1/2, a2 -> 1/5, a3 -> 2/5, a4 -> 4/3, a5 -> 17/15, a6 -> 1/4, b1 -> 1/2, b2 -> 2/5, b3 -> 11/20, b4 -> 1/2, b5 -> 11/10, b6 -> 1/2}, {x1 -> 0, y1 -> 3/2, r1 -> 1, r2 -> 1/4, k1 -> 3/4, k2 -> 0, a1 -> 1/2, a2 -> 1/5, a3 -> 2/5, a4 -> 0, a5 -> 3/5, a6 -> 1/4, b1 -> 1/2, b2 -> 2/5, b3 -> 11/20, b4 -> 1/2, b5 -> 7/10, b6 -> 1/2}}

Length[solr2]

8

Verifying

And @@@ (eqns /. solr2)

{True, True, True, True, True, True, True, True}