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  "body": "\n## Definition\n\nLet $\\Gamma_1$ and $\\Gamma_2$ be [[circles|060W5CBZP6]] in the Euclidean plane with distinct centres $c_1 \\ne c_2$ and radii $r_1, r_2$.\nFor a point $P$, its [[power|02E9VT1RVQ]] with respect to $\\Gamma_i$ is $\\pi_i(P) = \\norm{P - c_i}^2 - r_i^2$.\nThe *radical axis* of $\\Gamma_1$ and $\\Gamma_2$ is the set\n$$\\set{P : \\pi_1(P) = \\pi_2(P)}.$$\n\n## Remarks\n\nExpanding the two powers, the condition becomes\n$$2 \\ip{P}{c_2 - c_1} = \\norm{c_2}^2 - \\norm{c_1}^2 - r_2^2 + r_1^2.$$\nSince $c_2 - c_1 \\ne 0$, this is a [[line|0MP9BCF6DP]] perpendicular to $c_1 c_2$.\nFor three circles with non-collinear centres, the three radical axes meet in one point, the radical centre.\n\nExample: let $\\Gamma_1$ be $x^2 + y^2 = 1$ and $\\Gamma_2$ be $(x - 2)^2 + y^2 = 1$.\nThen $\\pi_1 = x^2 + y^2 - 1$ and $\\pi_2 = x^2 - 4x + 4 + y^2 - 1$, and they are equal exactly when $x = 1$.\n\nNon-example: concentric circles fail the condition $c_1 \\ne c_2$.\nFor $c_1 = c_2$ and $r_1 \\ne r_2$, the powers differ by the constant $r_1^2 - r_2^2 \\ne 0$, so no point has equal powers.\n",
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