Some theorems in euclidean geometry have incomplete proofs

Question

I have seen that, in euclidean geometry, proofs of some theorems use one instance of the 'geometric shape'(on which the theorem is based) to proof the theorem. Like, the proof of 'A straight line that divides any two sides of a triangle proportionally, is parallel to the third side' use only one instance of a triangle---like:

∆ABC is the instance

Then, constructions are added to this diagram to prove the theorem.

Clearly, the proof is not general.Because, only a triangle is in view. Therefore, this proof is not precise. We have had a general proof, must be having.I haven't yet visualized what the general proof might be. So, why do people call the above type of proof ,a proof?Is it a complete mathematical proof?

Answer 1

The diagram is meant to make it easier for you to explain and jot down facts. You cannot use "obvious properties" in the diagram to motivate your argument. For example, you cannot assume that all angles of a triangle must be acute.

It is possible for geometric proofs that are heavily based on diagrams to be wrong. A common example is that "All triangles are isosceles", in which the flaw lies in making an innocuous assumption about the position of a point.

Other instances include arguments about side lengths and angles. For example, if points $A< B, C$ are on a line, is it true that $|AC| = |AB|+|BC|$? In a general setting, this requires the use directed lengths.

Answer 2

I don't have a full answer, but here's a relevant observation about your example. The statement is actually a statement of affine geometry, since it deals only with proportions of parallel line segments, not absolute lengths or angles. In affine geometry, all triangles are equivalent, so we can consider any triangle without loss of generality.

With that information in hand, we are free to formulate a proof whose intermediate steps make restrictive assumptions about the triangle. For example, it might be convenient to assume that the altitude from $P$ to $\overleftrightarrow{AB}$ intersects $\overline{AB}$. This is true if the triangle is acute, so we will assume without loss of generality that it's acute! The resulting proof will superficially appear to apply to a proper subset of triangles, but the "w.l.o.g." step extends the result to all triangles.

Answer 3

To prove a general statement about all triangles, it suffices to consider one triangle and give a proof for that triangle provided one does not use any additional assumptions about that triangle. The reason this suffices is that, because no special assumptions were used, the same proof will apply to all other triangles equally well.

Answer 4

This is and is not a problem.

The proof is in the words, not the diagram. The diagram merely suggests which words to use, or illustrates steps in the proof that would otherwise be hard to follow. Once a proof is written down, the diagram is logically irrelevant and not part of the proof. In this sense the diagram cannot have any effect on the validity of the demonstration.

A proof inspired by a limited or inaccurate set of diagrams can overlook some possible cases or make invalid assumptions about the position of objects in the diagram. This is not as big a problem as it might appear to be, because proofs that are correct for the cases depicted by some (accurately drawn) diagrams, can often be interpreted in ways that cover the general case, such as allowing negative values for length, area and angle measurements.