Determination of KD from inhibition assay steady state data

Some definitions for the following considerations:
L=ligand
A=analyte (the ligand's binding partner. In a direct assay format, this is the compound for which KD should be determined; in the inhibition assay this is the recognition element binding to L as well as to C via the same binding site)
C=competitor (i.e. the compound for which KD should be determined in the inhibition assay; C competes with L for A)

For direct binding assays the situation is obviously pretty simple. For 50% ligand saturation (i.e. 50% Rmax) [L] equals [LA] and [A] equals [A]0 (since the solution of A is constantly replaced). Plugging in [L]=[LA] into the law of mass action directly results in KD=[A]0(50% Rmax). Thus the injected concentration of A leading to 50% saturation equals KD.

For an inhibition assay the Biacore concentration analysis handbook states that the concentration of C leading to 50% inhibition ([C]50) is given by:

[C]50 = KD + 1/2*[A]0

with KD the equilibrium dissociation constant for the interaction between A and C. Solving for KD yields:

KD = [C]50 - 1/2*[A]0

So far so good, but how can I derive this equation? And why is KD for the interaction between L and A cancelled out?

Thanks in advance for telling me your ideas!