Given:
Point P (x, y) is equidistant from points A (a, b) and B (a - b, a + b).

To prove:
bx = ay

Proof:

Step 1: Find the distance between point P and points A and B

The distance between two points (x1, y1) and (x2, y2) is given by the distance formula:

d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

The distance from point P to point A is:

d1 = sqrt((a - x)^2 + (b - y)^2)

The distance from point P to point B is:

d2 = sqrt((a - b - x)^2 + (a + b - y)^2)

Step 2: Equate d1 and d2

Since point P is equidistant from points A and B, we have:

d1 = d2

sqrt((a - x)^2 + (b - y)^2) = sqrt((a - b - x)^2 + (a + b - y)^2)

Squaring both sides:

(a - x)^2 + (b - y)^2 = (a - b - x)^2 + (a + b - y)^2

Step 3: Simplify the equation

Expanding the squares:

(a^2 - 2ax + x^2) + (b^2 - 2by + y^2) = (a^2 - 2ab - 2ax + 2bx + x^2) + (a^2 + 2ab - 2ay - 2by + y^2)

Cancelling out like terms:

a^2 + b^2 - 2ax - 2by = a^2 + 2ab - 2ay + 2bx

Step 4: Rearrange the equation

Grouping like terms:

b^2 - 2bx - 2by = 2ab - 2ax - 2ay + a^2 - a^2

Rearranging:

-2bx - 2by + 2ax + 2ay = 2ab - a^2

Step 5: Factor out common terms

Factoring out 2:

2(-bx - by + ax + ay) = 2(ab - a^2)

Step 6: Divide both sides by 2

Dividing both sides by 2:

-bx - by + ax + ay = ab - a^2

Step 7: Rearrange the equation

Rearranging:

ax + ay = ab - a^2 + bx + by

Step 8: Rearrange the equation again

Rearranging:

ay - by = ab - a^2 + bx - ax

Step 9: Factor out common terms

Factoring out y:

y(a - b) = a(b - a) + x(b - a)

Step 10: Rearrange the equation

Rearranging:

y(a - b) = -a(a -