Find the value of `(a^2+b^2+c^2)/R^2` in any right-angled triangle.

Explanation

To find the value of \(\frac{a^2 + b^2 + c^2}{R^2}\) in any right-angled triangle, we can follow these steps:

### Step 1: Identify the right triangle properties
In a right-angled triangle, we can denote the sides as follows:
- Let \(a\) be the length of the side opposite the right angle (hypotenuse).
- Let \(b\) and \(c\) be the lengths of the other two sides.

### Step 2: Apply the Pythagorean theorem
According to the Pythagorean theorem, we have:
\[ a^2 = b^2 + c^2 \]

### Step 3: Substitute into the expression
We need to find the expression \(\frac{a^2 + b^2 + c^2}{R^2}\). We can rewrite \(a^2 + b^2 + c^2\) using the Pythagorean theorem:
\[ a^2 + b^2 + c^2 = (b^2 + c^2) + b^2 + c^2 = 2b^2 + 2c^2 \]
However, we can also express it as:
\[ a^2 + b^2 + c^2 = a^2 + (b^2 + c^2) = a^2 + a^2 = 2a^2 \]

### Step 4: Use the circumradius formula
The circumradius \(R\) of a right triangle can be expressed as:
\[ R = \frac{a}{2} \]
Thus, \(R^2 = \left(\frac{a}{2}\right)^2 = \frac{a^2}{4}\).

### Step 5: Substitute \(R^2\) into the expression
Now we substitute \(R^2\) into our expression:
\[ \frac{a^2 + b^2 + c^2}{R^2} = \frac{2a^2}{\frac{a^2}{4}} = 2a^2 \cdot \frac{4}{a^2} = 8 \]

### Final Result
Therefore, the value of \(\frac{a^2 + b^2 + c^2}{R^2}\) in any right-angled triangle is:
\[ \boxed{8} \]