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📅 May 14, 2026 👤 scraface

May 14, 2026

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📌 c = \frac{a + b - z}{2}

📌 The area of the triangle is $A = \frac{1}{2}ab$, and the area of the inscribed circle is $\pi c^2$. The ratio is:

📌 \frac{\pi c^2}{\frac{1}{2}ab}

📌 From the identity $a + b = 2c + z$, and using the Pythagorean theorem $a^2 + b^2 = z^2$, we aim to express $ab$ in terms of $z$ and $c$. Note:

📌 Thus, the ratio becomes:

📌 \frac{\pi c^2}{\frac{1}{2}(2c^2 + 2cz)} = \frac{\pi c^2}{c^2 + cz} = \frac{\pi c}{c + z}

📌 So the ratio of the area of the circle to the area of the triangle is:

📌 \boxed{\frac{\pi c}{c + z}}

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