Geometric False Result Based on SSA Congruence Criterion for Triangles I'm writing an article to warn students not to use SSA (Side-Side-Angle) criterion to prove congruence between two triangles. I'd like to add to the article an example of a geometric fallacy based on SSA. Do you know any such fallacy?
 A: 
Here is an example with $|AC|=1,|BC|=\sqrt{2},\text{ and }\angle ABC=30^\circ$
Alternate version of same diagram. Obviously the two triangles are not congruent even though they share two sides and an angle.

A: I tried create a fallacy. Maybe it is not totally right or it is not exactly what you want. But, maybe, it give you some idea.
Look the picture below.

By construction, 
$$\text{diameter of the blue circle}>\text{length of the black segment}+\text{length of the orange segment}$$
Furthermore:


*

*The red angles have tha same measure (by symmetry).

*The blue line segments have the same length (because they are radius of the same circle).

*The green line segments has the same length (because the green circles have the same radius).
It follows from the SSA criterion that the black and orange line segments have the same length. So,
\begin{align}
\text{diameter of the blue circle}&>\text{length of the black segment}+\text{length of the orange segment}\\
&=\text{length of the orange segment}+\text{length of the orange segment}\\
&=2\times \text{radius of the blue circle}
\end{align}
and thus, the formula
$$\text{diameter}=2\times\text{radius}$$
is not valid in general.
