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📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 Let $r = \sqrt{x^2 + y^2}$. Then:

📌 r^2 + z^2 = 4r \Rightarrow r^2 - 4r + z^2 = 0 \Rightarrow (r - 2)^2 + z^2 = 4

📌 This is a sphere of radius 2 centered at $(2, 0, 0)$ in cylindrical coordinates, which corresponds to a **sphere tangent to the y-z plane** in Cartesian coordinates.

📌 oxed{(x - 2)^2 + y^2 + z^2 = 4}

📌 Question: A regular tetrahedron has three of its vertices at $A = (1, 1, 1)$, $B = (1, -1, -1)$, and $C = (-1, 1, -1)$. Find the integer coordinates of the fourth vertex $D$, assuming all coordinates are integers.

📌 Solution: In a regular tetrahedron, all edges are equal. Compute distances:

📌 AB = \sqrt{(1 - 1)^2 + (1 + 1)^2 + (1 + 1)^2} = \sqrt{0 + 4 + 4} = \sqrt{8} \

📌 AC = \sqrt{(1 + 1)^2 + (1 - 1)^2 + (1 + 1)^2} = \sqrt{4 + 0 + 4} = \sqrt{8} \

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