How to get steps of the solutions for:
Solve[x^2 +
y^2 == (r - 65)^2 && (x - 85)^2 + (y + 185)^2 == (r -
55)^2 && (x - 150)^2 + (y + 45)^2 == (r - 35)^2, {x, y, r}]
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2 Answers
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One of ways is as follows. We start from
Expand[x^2 + y^2 == (r - 65)^2 && (x - 85)^2 + (y + 185)^2 == (r -
55)^2 && (x - 150)^2 + (y + 45)^2 == (r - 35)^2]
x^2 + y^2 == 4225 - 130 r + r^2 && 41450 - 170 x + x^2 + 370 y + y^2 == 3025 - 110 r + r^2 && 24525 - 300 x + x^2 + 90 y + y^2 == 1225 - 70 r + r^2
We see x^2+y^2 in the second and third equations. We simplify those making use of the first equation.
Simplify[ 41450 - 170 x + x^2 + 370 y + y^2 == 3025 - 110 r + r^2 &&
24525 - 300 x + x^2 + 90 y + y^2 == 1225 - 70 r + r^2,
Assumptions -> x^2 + y^2 == 4225 - 130 r + r^2]
2 r + 17 x == 4265 + 37 y && 4 (r + 5 x) == 1835 + 6 y
Now we can express x and y through r by solving a system of linear equations
Solve[2 r + 17 x == 4265 + 37 y && 4 (r + 5 x) == 1835 + 6 y, {x, y}]
{x -> 1/638 (42305 - 136 r), y -> 1/638 (-54105 - 28 r)}}
At last, we substitute the above expressions in the first equation and solve it in r.
x^2 + y^2 == (r - 65)^2 /. {x -> 1/638 (42305 - 136 r), y -> 1/638 (-54105 - 28 r)}
(42305 - 136 r)^2/407044 + (-54105 - 28 r)^2/407044 == (-65 + r)^2
Solve[(42305 - 136 r)^2/407044 + (-54105 - 28 r)^2/407044 == (-65 + r)^2, {r}, Reals]
{{r -> (11109660 - 1595 Sqrt[162728790])/193882}, {r -> ( 11109660 + 1595 Sqrt[162728790])/193882}}
The rest is left on your own.
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Reduce on the equations works with back substitution:
eq = {x^2 +
y^2 == (r - 65)^2, (x - 85)^2 + (y + 185)^2 == (r -
55)^2, (x - 150)^2 + (y + 45)^2 == (r - 35)^2};
ans = {ToRules@Reduce[eq, {x, y, r}]};
a1 = ({x, y, r} /. ans[[1, {2, 3}]]) /. ans[[1, 1]]
a2 = ({x, y, r} /. ans[[2, {2, 3}]]) /. ans[[2, 1]]
N /@ {a1, a2}
Illustrating the solutions:
ctr = {{0, 0}, {85, -185}, {150, -45}};
ContourPlot[
Evaluate[eq /. r -> N[a1[[3]]]], {x, -300, 300}, {y, -300, 300},
Epilog -> {Red, PointSize[0.02], Point[a1[[{1, 2}]]], Black,
Point[ctr]}, PlotLegends -> "Expressions"]
ContourPlot[
Evaluate[eq /. r -> N[a2[[3]]]], {x, -300, 300}, {y, -300, 300},
Epilog -> {Red, PointSize[0.02], Point[a2[[{1, 2}]]], Black,
Point[ctr]}, PlotLegends -> "Expressions"]
