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Home » Perhaps typo in setup? But assume exact: no integer solution. - Midis

Perhaps typo in setup? But assume exact: no integer solution. - Midis

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 بتوسيع وتعويض $x^2 + y^2 + z^2 = 2a^2$:

📌 x^2 - 2ax + a^2 + y^2 - 2ay + a^2 + z^2 = 2a^2 \quad \Rightarrow \quad 2a^2 - 2ax - 2ay + 2a^2 = 2a^2 \quad \Rightarrow \quad 2a^2 - 2ax - 2ay + 2a^2 - 2a^2 = 0 \quad \Rightarrow \quad ax + ay = a^2 \quad \Rightarrow \quad x + y = a

📌 أخيرًا، المسافة من $D$ إلى $C$:

📌 \sqrt{x^2 + (y-a)^2 + (z-a)^2} = a\sqrt{2} \quad \Rightarrow \quad x^2 + (y-a)^2 + (z-a)^2 = 2a^2

📌 x^2 + y^2 - 2ay + a^2 + z^2 - 2az + a^2 = 2a^2 \quad \Rightarrow \quad 2a^2 - 2ay - 2az + 2a^2 = 2a^2 \quad \Rightarrow \quad 2a^2 - 2ay - 2az + 2a^2 - 2a^2 = 0 \quad \Rightarrow \quad ay + az = a^2 \quad \Rightarrow \quad y + z = a

📌 x + y = a \quad \text{و} \quad y + z = a

📌 نعوض $y = a - x$ في $y + z = a$:

📌 a - x + z = a \quad \Rightarrow \quad z = x

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