Peaucellier-Lipkin Cell

Move the blue point.  What is the value of the product OP*OP' ?

b

b

b

a

Hint

Result

a_1 = 9 (slider from 0 to 20, step 0.1)
b_1 = 7 (slider from 0 to 15, step 0.1)
A_1 = (2.4, 3.66)
c = Circle(A_1, a_1)
B_1 = (4.86, 4.84)
d = Circle(B_1, b_1)
C, D = Intersect(d, c)
E = Intersect(d, c, 2)
e = Line(D, C)
B' = Mirror(B_1, e)
a = Segment(A_1, C)
g = Segment(C, B')
h = Segment(B', D)
i = Segment(D, B_1)
b = Segment(B_1, C)
k = Segment(A_1, D)
A = Mirror(C, e)
B = Mirror(D, e)
P' = Intersect(g, h)
P = Intersect(b, i)
O = Intersect(a, k)
l = false
m = Segment(C, D)
n = Segment(B', A_1)
F = Intersect(m, n)
M = Intersect(m, n)
OP' = Distance(O, P')
OP = Distance(O, P)
OPOP': OP * OP'
OM = Distance(O, M)
PM = Distance(P, M)
MP' = Distance(M, P')
OMPM: OM * PM
OMMP': OM * MP'
OM2: OM * OM
MP = Distance(M, P)
MP2: MP * MP
sqrtOPOP': sqrt(OPOP')
OA = Distance(O, A)
aDist = Distance(O, A)
bDist = Distance(P, A)
text1: "OP × OP' = " + (LaTeX[OP]) + " × " + (LaTeX[OP']) + " = " + (LaTeX[OPOP']) + " =  (" + (LaTeX[sqrtOPOP']) + ")^2 = (" + (LaTeX[aDist]) + ")^2 - (" + (LaTeX[bDist]) + ")^2 = a^2-b^2"
f = false