Geometry help

[rain tiger]

Could i get some help with this question please? Have not seen one like this before

[left slate]

Is it right?

[agile prawn]

AP = h1
PD = h2
AD = h1 + h2
S(ABCD) = (1/2)(x + y)(h1 + h2)
S(ABQP) = (1/2)(x + r)h1
S(PQCD) = (1/2)(r + y)h2

S(ABCD) = S(ABQP) + S(PQCD)
(1/2)(x + y)(h1 + h2) = (1/2)(x + r)h1 + (1/2)(r + y)h2
(1/2)xh1 + (1/2)xh2 + (1/2)yh1 + (1/2)yh2
= (1/2)xh1 + (1/2)rh1 + (1/2)rh2 + (1/2)yh2
(1/2)xh2 + (1/2)yh1 = (1/2)rh1 + (1/2)rh2
xh2 + yh1 = rh1 + rh2
yh1 - rh1 = rh2 - xh2
h1(y - r) = h2(r - x)
h1/h2 = (r - x)/(y - r) -> (1)
S(ABQP) = S(PQCD)
(1/2)(x + r)h1 = (1/2)(y + r)h2
h1(x + r) = h2(y + r)
xh1 + rh1 = yh2 + rh2
h1(x + r) = h2(y + r)
h1/h2 = (y + r)/(x + r) -> (2)
(1) = (2)
(r - x)/(y - r) = (y + r)/(x + r)
r² - x² = y² - r²
x² + y² = 2r²

[left slate]

Is my method right?