Statements in Euclidean geometry that appear to be true but aren't

I'm teaching a geometry course this semester, involving mainly Euclidean geometry and introducing non-Euclidean geometry. In discussing the importance of deductive proof, I'd like to present some examples of statements that may appear to be true (perhaps based on a common student misconception or over-generalisation), but are not. The aim would be to reinforce one of the reasons given for studying deductive proof: to properly determine the validity of statements claimed to be true.

Can anyone offer interesting examples of such statements?

An example would be that the circumcentre of a triangle lies inside the triangle. This is true for triangles without any obtuse angles - which seems to be the standard student (mis)conception of a triangle. However, I don't think that this is a particularly good example because the misconception is fairly easily revealed, as would statements that hold only for isoceles or right-angled triangles. I'd really like to have some statements whose lack of general validity is quite hard to tease out, or has some subtlety behind it.

Of course the phrase 'may appear to be true' is subjective. The example quoted should be understood as indicating the level of thinking of the relevant students.

Thanks.

Also trivial but maybe interesting: two sides and an angle determine a triangle up to congruence.

Here is one example that is quite similar in nature to the statement in the question about the center of the circumcircle lying inside a triangle, but the dubious part ("lie inside") is somewhat better disguised. I report it only because I just found it in Wikipedia, with a literature reference.

The incenter (that is, the center for the inscribed circle) of the orthic triangle is the orthocenter of the original triangle.

It is easily verified that for a triangle with an obtuse angle the orthocenter lies outside the orthic triangle, so it cannot be the incenter in this case; it is one of the excenters instead.

A common attempt to trisect an angle is to construct an isosceles triangle, $\triangle ABC$, with $\angle ABC$ the desired angle; then trisect the opposite side $\overline{AC}$, finding points $D,E$ on $\overline{AC}$ with $AD = DE = EC$. One might guess that $m\angle ABD = m\angle DBE = m\angle EBC$ but in fact this is never the case.

Although this isn't quite what you were asking for, the best example of a subtly flawed geometric proof I know of can be seen at Wikipedia here.