Let the fixed vertex be P = (20,20). Translate so P = (0,0).
Let the other vertices be u = (a,b), v = (c,d).
Area condition:
(1/2)|ad - bc| = 9 => |ad - bc| = 18
CASE 1: Equal sides meet at P
|u| = |v|
=> |u|^2 = |v|^2 = n, and u.v = m
Use identity:
(u.v)^2 + (ad - bc)^2 = |u|^2 * |v|^2
m^2 + 18^2 = n^2
=> n^2 - m^2 = 324
=> (n - m)(n + m) = 324
Valid n that are sums of two squares:
n = 18, 82
n = 18: (±3, ±3) → 4 triangles
n = 82: (±9, ±1), (±1, ±9) → 8 triangles
Total = 12
CASE 2: Equal sides meet at another vertex
|u| = |u - v|
=> |v|^2 = 2(u.v)
Let |u|^2 = x, |v|^2 = y
xy - (y/2)^2 = 324
=> 4xy - y^2 = 1296
=> y(4x - y) = 1296
Valid solutions (both sums of two squares):
(x, y) = (82, 4): v = (±2,0), (0,±2) → 8 triangles
(x, y) = (18, 36): v = (±6,0), (0,±6) → 8 triangles
Total = 16
FINAL ANSWER:
12 + 16 = 28