$20 \cdot 87$ As shown in the figure, in a semicircle with diameter $AB$, there is an inscribed square $CDEF$ with a side length of 1. If $AC=a, BC=b$. Then which of the following relationships is incorrect?
(A) $a-b=1$.
(B) $a b=1$.
(C) $a+b=\sqrt{5}$.
(D) $a^{2}+b^{2}=5$.
[Solution] By symmetry, we have $A F = C B$, so $a - b = 1$ holds, meaning (A) is true. Also, by the intersecting chords theorem, we have $a b = C D^2 = 1^2 = 1$, meaning (B) is true.
Substituting $b = \frac{1}{a}$ into equation (A), we get

that is,
$$
\begin{array}{c}
a - \frac{1}{a} = 1, \\
a^2 - a - 1 = 0,
\end{array}
$$

Solving this, we get
$$
a = \frac{1 \pm \sqrt{5}}{2} \text{ (discard the negative value). }
$$

Thus,
$$
a + b = a + (a - 1) = 2a - 1 = \sqrt{5},
$$

indicating that (C) is true.
In summary, (A), (B), and (C) are all true.
Therefore, the answer is (D).