If 2s = a + b + c, then what is s² + (s−a)(s−b) + (s−b)(s−c) + (s−c)(s−a) equal to?

Let x=s−a, y=s−b, z=s−c. Then x+y+z = 3s−(a+b+c) = s. So s² = (x+y+z)². And xy+yz+zx terms. Expand: s²+xy+yz+zx = (x+y+z)²+xy+yz+zx = x²+y²+z²+3(xy+yz+zx). Hmm complicated. Direct: try a=b=c=1, s=3/2. s²=9/4. (s−a)=1/2 each, products = 1/4 each, sum = 3/4. Total = 9/4+3/4 = 3. ab+bc+ca = 3. So answer ab+bc+ca = 3 matches. ✓