A Geometric Identity
Author: John Maharry
Date: 4/25/2007

Click on the applet to activate it. 
Adjust the dimensions of the oval by moving the "sliders".
Move the position of the point "N" by sliding the angle.

Notice that the Identity is always true.

Can you find a geometric proof?

This problem came from a publication of the AMS.

α = 0.7679448708775051 rad (slider from 0.017453292519943295 to 1.5533430342749532, step 0.017453292519943295)
a = 6.7 (slider from 0 to 10, step 0.1)
b = 3 (slider from 0 to 5, step 0.1)
B: (0, b)
M: (0, -y(B))
A: (a, 0)
K: (-x(A), 0)
N: (cos(α) a, sin(α) b)
d = Conic(B, K, A, M, N)
c: 1 - a²
e = Tangent(N, d)
Q = Intersect(e, xAxis)
f = OrthogonalLine(N, e)
P = Intersect(f, xAxis)
C = Intersect(xAxis, yAxis)
β = Angle(P, C, N)
g = Segment(C, N)
T1: "A² + B² =   " + (Distance[A, C]² + " + " + (Distance[B, C]²)) + " = " + (Distance[A, C]² + Distance[B, C]²)
T2: " NP x NQ = " + (Distance[N, P]) + " x " + (Distance[N, Q]) + " = " + (Distance[N, P] Distance[N, Q])