Two circles in the first quadrant of radii $$r_1$$ and $$r_2$$ touch the coordinate axes. Each of them cuts off an intercept of 2 units with the line $$x + y = 2$$. Then $$r_1^2 + r_2^2 - r_1 r_2$$ is equal to _____.

We need to find $$r_1^2 + r_2^2 - r_1 r_2$$ for two circles in the first quadrant that touch both coordinate axes and each cuts an intercept of 2 units on the line $$x + y = 2$$.

A circle of this type has center $$(r,r)$$ and radius $$r$$.

$$ (x - r)^2 + (y - r)^2 = r^2 $$

In order to determine the chord length cut by the line, first compute the perpendicular distance from the center to that line. For the center $$(r,r)$$ and the line $$x + y - 2 = 0$$, this distance is

$$ d = \frac{|r + r - 2|}{\sqrt{2}} = \frac{|2r - 2|}{\sqrt{2}} $$

Since the intercept (or chord) length on the line is 2 units, the chord length formula $$2\sqrt{r^2 - d^2} = 2$$ implies that

$$ r^2 - d^2 = 1 $$

Substituting the value of $$d^2 = \frac{(2r-2)^2}{2} = \frac{4r^2 - 8r + 4}{2} = 2r^2 - 4r + 2$$ into this equation yields

$$ r^2 - (2r^2 - 4r + 2) = 1 $$

$$ r^2 - 2r^2 + 4r - 2 = 1 $$

$$ -r^2 + 4r - 3 = 0 $$

$$ r^2 - 4r + 3 = 0 $$

Factoring the quadratic expression gives

$$ (r - 1)(r - 3) = 0 $$

Accordingly, the two possible radii are $$r_1 = 1$$ and $$r_2 = 3$$.

Next, computing the desired combination yields

$$ r_1^2 + r_2^2 - r_1 r_2 = 1 + 9 - 3 = 7 $$

Therefore, the required value is $$7$$.