# Revision history [back]

Well, as this is a purely polynomial problem (by replacing $\tan(t$) by $t$) of intersection between two circles, you can use polynomial tools.

sage: x,y,d,t,h,k,r = polygen(QQ,'x,y,d,t,h,k,r')
sage: eq = [x*x+y*y-d*d,y-t*x,(x-h)**2+(y-k)**2-r**2,h-1,k-1,r-6,t-2]
sage: ring = x.parent()
sage: ideal = ring.ideal(eq)
sage: ideal.dimension()
0
sage: ideal.variety()   # no solutions with rational coordinates
[]
sage: ideal.variety?   # help of the variety method
sage: ideal.variety(ring=CC)     # working over the complex numbers
[{y: 6.55163526410386, r: 6.00000000000000, t: 2.00000000000000, d: -7.32495090716051, k: 1.00000000000000, h: 1.00000000000000, x: 3.27581763205193}, {y: -4.15163526410386, r: 6.00000000000000, t: 2.00000000000000, d: -4.64166933416076, k: 1.00000000000000, h: 1.00000000000000, x: -2.07581763205193}, {y: -4.15163526410386, r: 6.00000000000000, t: 2.00000000000000, d: 4.64166933416076, k: 1.00000000000000, h: 1.00000000000000, x: -2.07581763205193}, {y: 6.55163526410386, r: 6.00000000000000, t: 2.00000000000000, d: 7.32495090716051, k: 1.00000000000000, h: 1.00000000000000, x: 3.27581763205193}]
sage: ideal.variety(ring=RR)     # working over the real numbers
[{y: 6.55163526410386, r: 6.00000000000000, t: 2.00000000000000, d: -7.32495090716051, k: 1.00000000000000, h: 1.00000000000000, x: 3.27581763205193}, {y: -4.15163526410386, r: 6.00000000000000, t: 2.00000000000000, d: -4.64166933416076, k: 1.00000000000000, h: 1.00000000000000, x: -2.07581763205193}, {y: -4.15163526410386, r: 6.00000000000000, t: 2.00000000000000, d: 4.64166933416076, k: 1.00000000000000, h: 1.00000000000000, x: -2.07581763205193}, {y: 6.55163526410386, r: 6.00000000000000, t: 2.00000000000000, d: 7.32495090716051, k: 1.00000000000000, h: 1.00000000000000, x: 3.27581763205193}]