My numbers are 20, 99, and 101, which fit a Pythagorean triple.

20^2+ 99^2= 101^2

However, where do these fit into Euclid's formula? A = 2mn, B = m^2- n^2, and C = m^2+ n^2.

What are m and n?

if you look at C-B what do you notice?
m^2 +n^2 - (m^2 -n^2) (combine like terms)
m^2-m^2 + n^2+n^2
0m^2 +2n^2
so C-B = 2(n^2)
how far apart are two of the three numbers?
101-99 = 2.
so if n were 1, then C-B = 2.
amazing isn't it?
now that we know n is 1,
plug in the numbers
A=2m(1)
A=2m
which number have we not used out of our three numbers?
must be the 20
20 = 2(m)
20/2 = 2m/2
10 = m
now we know m,
plug in the numbers
B= 10^2 - 1^2
C= 10^2 + 1^2

to review:
Euclid's formula? A = 2mn, B = m^2- n^2, and C = m^2+ n^2.
What are m and n?

But where are you getting 20 from? When you say "which number have we not used out of our three numbers?" I don't see that 20 was ever mentioned..