Solution: For a right triangle with legs $ a $ and $ b $, and hypotenuse $ c $, the radius $ r $ of the inscribed circle is given by:

["Understanding the Formula: Radius of the Inscribed Circle in a Right Triangle", "In geometry, a right triangle with legs ( a ) and ( b ), and hypotenuse ( c ), offers a rich opportunity to explore elegant formulas—particularly the radius ( r ) of its inscribed circle. This radius, often overlooked, plays a crucial role in both theoretical mathematics and practical applications such as architectural design, engineering, and physics.", "What is the Radius ( r ) of the Inscribed Circle?", "For any triangle, the radius ( r ) of the inscribed circle (incircle) can be calculated using the formula:\n[\nr = \frac{a + b - c}{2}\n]\nHowever, this expression can also be elegantly written in terms of the area ( A ) and semi-perimeter ( s ) of the triangle, where:\n[\nr = \frac{A}{s}\n]\nFor a right triangle, this formula simplifies beautifully due to the Pythagorean Theorem: ( c = \sqrt{a^2 + b^2} ).", "Deriving the Formula for Right Triangles", "Let’s explore how the radius ( r = \frac{a + b - c}{2} ) emerges.", "1. Semi-perimeter:\n The semi-perimeter ( s ) of the triangle is:\n [\n s = \frac{a + b + c}{2}\n ]", "2. Area:\n The area ( A ) of the right triangle is:\n [\n A = \frac{1}{2}ab\n ]", "3. Inradius Formula:\n Using ( r = \frac{A}{s} ):\n [\n r = \frac{\frac{1}{2}ab}{\frac{a + b + c}{2}} = \frac{ab}{a + b + c}\n ]", "4. Equating the Two Expressions\n Through algebraic manipulation and substitution using ( c = \sqrt{a^2 + b^2} ), it can be shown that:\n [\n \frac{a + b - c}{2} = \frac{ab}{a + b + c}\n ]\n This equality confirms that both expressions describe the radius of the incircle for a right triangle, each offering unique insights.", "Why This Formula Matters", "Understanding ( r = \frac{a + b - c}{2} ) helps students and professionals alike in several ways:", "- Efficiency: Using this direct formula requires only the lengths of the two legs, avoiding area calculations.\n- Geometric Insight: The formula reveals a deep connection between the triangle’s side lengths and its incircle.\n- Applications: From optimizing space in design to analyzing electric circuits involving triangular resistance models, this radius aids diverse real-world problem solving.", "Conclusion", "The formula for the radius ( r ) of the inscribed circle in a right triangle —\n[\nr = \frac{a + b - c}{2}\n]\n— stands as a remarkable example of how geometry blends simplicity and power. Whether for mathematical exploration, physics modeling, or engineering challenges, mastering this formula unlocks a deeper understanding of triangles and their circles.", "---", "Key Takeaways:\n- Radius of incircle: ( r = \frac{a + b - c}{2} )\n- Based on semi-perimeter and area formulas\n- Simplifies calculations for right triangles\n- Essential for both pure and applied mathematics", "---", "Keywords: inscribed circle radius, incircle formula, right triangle, a and b legs, hypotenuse c, geometric formulas, a + b - c / 2, triangle incircle, math education, geometry applications\nRead more about triangle geometry and circle theorems in math resources and textbooks."]