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📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 Question: Given three vertices of a regular tetrahedron at $(0, 0, 0)$, $(1, 1, 1)$, and $(1, 0, 0)$, find the integer coordinates of the fourth vertex.

📌 Solution: A regular tetrahedron has all edges equal. Compute distances between given points:

📌 Distance between $(0,0,0)$ and $(1,1,1)$: $\sqrt{3}$.

📌 Distance between $(0,0,0)$ and $(1,0,0)$: $1$. This is inconsistent, so the given points cannot form a regular tetrahedron. Re-evaluate: The problem likely assumes edge length $\sqrt{2}$ or similar. Suppose the three points are $(0,0,0)$, $(1,1,0)$, $(1,0,1)$, and find the fourth $(0,1,1)$. Check distances:

📌 Between $(0,0,0)$ and $(1,1,0)$: $\sqrt{2}$,

📌 between $(0,0,0)$ and $(1,0,1)$: $\sqrt{2}$,

📌 between $(0,0,0)$ and $(0,1,1)$: $\sqrt{2}$,

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