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Home » Tropicalement Addicted: Discover the Secret to Echoing Paradise Every Day! - Abu Waleed Tea

Tropicalement Addicted: Discover the Secret to Echoing Paradise Every Day! - Abu Waleed Tea

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 ho\sin\phi = \sqrt{x^2 + y^2}

📌 x^2 + y^2 + z^2 = 4\sqrt{x^2 + y^2}

📌 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:

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