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  "body": "\n## Proof\n\nWe show that each product equals $r^2 - d(S, O)^2$, where $O$ is the centre and $r$ the radius.\n\n1. The points are [[concyclic|049DCJZ448]] in $\\mathbb{R}^2$. So there are a point $O$ and a number $r \\geq 0$ with $d(X, O) = r$ for $X = A, B, C, D$, where $d$ is the [[Euclidean distance|0NPPGHQNXQ]].\n2. Put $q(t) = t^2 + 2t\\, u \\cdot (S - O) + \\norm{S - O}^2 - r^2$ for a unit vector $u$. With the [[dot product|0S52TNNTH9]], $\\norm{S + tu - O}^2 - r^2 = q(t)$. So $S + tu$ is at distance $r$ from $O$ if and only if $q(t) = 0$.\n3. Claim: let $X$, $Y$ be points at distance $r$ from $O$, and let $S = (1 - s)X + sY$ with $0 \u003c s \u003c 1$. Then $d(X, S)\\, d(S, Y) = r^2 - \\norm{S - O}^2$.\n4. Proof of the claim when $X = Y$: then $S = X$, so both sides are $0$.\n5. Now let $X \\neq Y$, and let $u = (Y - X)/\\norm{Y - X}$ and $L = \\norm{Y - X}$. Then $X = S + t_1 u$ and $Y = S + t_2 u$ with $t_1 = -sL \u003c 0$ and $t_2 = (1 - s)L \u003e 0$.\n6. By step 2, $q(t_1) = q(t_2) = 0$. Write $q(t) = t^2 + pt + k$ with $k = \\norm{S - O}^2 - r^2$.\n7. Subtracting, $0 = q(t_1) - q(t_2) = (t_1 - t_2)(t_1 + t_2 + p)$. Since $t_1 \\neq t_2$, we get $p = -(t_1 + t_2)$.\n8. Then $0 = q(t_1) = t_1^2 - (t_1 + t_2)t_1 + k = k - t_1 t_2$. So $t_1 t_2 = \\norm{S - O}^2 - r^2$.\n9. Since $u$ is a unit vector, $d(X, S) = \\abs{t_1}$ and $d(S, Y) = \\abs{t_2}$. By step 5, $\\abs{t_1}\\abs{t_2} = -t_1 t_2$. By step 8, this is $r^2 - \\norm{S - O}^2$, which proves the claim.\n10. The point $S$ is inside the [[segment|VQ4Q2V45RP]] $AC$, so $S = (1 - s)A + sC$ with $0 \u003c s \u003c 1$. By the claim, $d(A, S)\\, d(S, C) = r^2 - d(S, O)^2$.\n11. In the same way, $S$ is inside the segment $BD$, so $d(B, S)\\, d(S, D) = r^2 - d(S, O)^2$.\n12. Hence $d(A, S)\\, d(S, C) = d(B, S)\\, d(S, D)$. Both products are minus the [[power|02E9VT1RVQ]] of $S$.\n\n## Idea\n\nAlong a line through $S$, the points of the [[circle|060W5CBZP6]] are the roots of a quadratic equation in the signed distance from $S$. The product of the roots is the constant term $d(S, O)^2 - r^2$, which does not depend on the line. The two roots have opposite signs, so each chord through $S$ gives the product $r^2 - d(S, O)^2$.\n",
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