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  "body": "\n## Proof\n\nWe show that each product equals the absolute value of $d(P, O)^2 - r^2$.\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. For a unit vector $u$, put $q(t) = t^2 + 2t\\, u \\cdot (P - O) + k$ with $k = \\norm{P - O}^2 - r^2$. With the [[dot product|0S52TNNTH9]], $\\norm{P + tu - O}^2 - r^2 = q(t)$.\n3. Claim: let $X \\neq Y$ be points at distance $r$ from $O$, and let $P$ be on the [[line|0MP9BCF6DP]] $XY$. Then $d(P, X)\\, d(P, Y) = \\abs{k}$.\n4. Proof of the claim: let $u = (Y - X)/\\norm{Y - X}$. The line $XY$ is $\\set{P + tu : t \\in \\mathbb{R}}$, since it contains $P$ and has the direction $u$. So $X = P + t_1 u$ and $Y = P + t_2 u$, with $t_1 \\neq t_2$ since $X \\neq Y$.\n5. By step 2, $q(t_1) = q(t_2) = 0$. Write $q(t) = t^2 + pt + k$.\n6. 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 $p = -(t_1 + t_2)$.\n7. Then $0 = q(t_1) = t_1^2 - (t_1 + t_2)t_1 + k = k - t_1 t_2$. So $t_1 t_2 = k$.\n8. Since $u$ is a unit vector, $d(P, X) = \\abs{t_1}$ and $d(P, Y) = \\abs{t_2}$. So $d(P, X)\\, d(P, Y) = \\abs{t_1 t_2} = \\abs{k}$, which proves the claim.\n9. Apply the claim to $X = A$, $Y = D$, which is allowed since $A \\neq D$ and $P$ is on the line $AD$. This gives $d(P, A)\\, d(P, D) = \\abs{k}$.\n10. Apply it to $X = B$, $Y = C$, since $B \\neq C$ and $P$ is on the line $BC$. This gives $d(P, B)\\, d(P, C) = \\abs{k}$.\n11. Hence $d(P, A)\\, d(P, D) = d(P, B)\\, d(P, C)$. Since $P$ is outside the [[circle|060W5CBZP6]], $k \u003e 0$, and both products equal the [[power|02E9VT1RVQ]] $k$ of $P$.\n\n## Idea\n\nAlong a line through $P$, the points of the circle are the roots of a quadratic equation in the signed distance from $P$. The product of the two roots is the constant term $d(P, O)^2 - r^2$. This term depends only on $P$, not on the line, so both secants give the same product.\n",
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