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  "body": "\n## Proof\n\nWe show that both sides equal the [[power|02E9VT1RVQ]] $k = d(P, O)^2 - r^2$ of $P$.\n\n1. Let $O$ be the centre and $r \u003e 0$ the radius of the [[circle|060W5CBZP6]] $C$, and let $d$ be the [[Euclidean distance|0NPPGHQNXQ]]. Put $k = \\norm{P - O}^2 - r^2$.\n2. For a unit vector $u$, put $q(t) = t^2 + pt + k$ with $p = 2\\, u \\cdot (P - O)$. With the [[dot product|0S52TNNTH9]], $\\norm{P + tu - O}^2 - r^2 = q(t)$. So $P + tu \\in C$ if and only if $q(t) = 0$.\n3. Since $P$ is outside $C$, $P \\neq T$. Let $u = (T - P)/\\norm{T - P}$ and $t_0 = \\norm{T - P} = d(P, T)$.\n4. The tangent line $\\ell$ contains $P$ and $T$, so it is the [[line|0MP9BCF6DP]] $\\set{P + tu : t \\in \\mathbb{R}}$. Also $T = P + t_0 u$, so $q(t_0) = 0$.\n5. Put $t_1 = -p - t_0$, so $p = -(t_0 + t_1)$. From $q(t_0) = 0$ we get $k = -t_0^2 - pt_0 = t_0 t_1$.\n6. Then $q(t) = t^2 - (t_0 + t_1)t + t_0 t_1 = (t - t_0)(t - t_1)$, so $q(t_1) = 0$. Hence $P + t_1 u \\in \\ell \\cap C$.\n7. Since $\\ell$ is [[tangent|0T7DX8NY1X]] at $T$, $\\ell \\cap C = \\set{T}$. So $P + t_1 u = T = P + t_0 u$, and $t_1 = t_0$.\n8. By steps 5 and 7, $k = t_0^2 = d(P, T)^2$.\n9. Now let $u'$ be a unit direction vector of the second line, so this line is $\\set{P + tu'}$. Then $G_1 = P + s_1 u'$ and $G_2 = P + s_2 u'$, with $s_1 \\neq s_2$ since $G_1 \\neq G_2$.\n10. Let $q'(t) = t^2 + p't + k$ with $p' = 2\\, u' \\cdot (P - O)$. By step 2 for $u'$, $q'(s_1) = q'(s_2) = 0$.\n11. Subtracting, $0 = (s_1 - s_2)(s_1 + s_2 + p')$, so $p' = -(s_1 + s_2)$. Then $0 = q'(s_1) = k - s_1 s_2$, so $s_1 s_2 = k$.\n12. Since $u'$ is a unit vector, $d(P, G_1)\\, d(P, G_2) = \\abs{s_1}\\abs{s_2} = \\abs{k}$.\n13. By step 8, $k = d(P, T)^2 \\geq 0$, so $\\abs{k} = k$. Hence $d(P, T)^2 = d(P, G_1)\\, d(P, G_2)$.\n\n## Idea\n\nOn each line through $P$, the points of the circle are the roots of a quadratic equation with the constant term $k$. For the tangent the two roots coincide, so $k$ is the square of the tangent length. For the secant, $k$ is the product of the two distances.\n",
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