Theorems, axioms, definitions
Proof. Theorem. Axiom. Initial notions. Definitions.
Proof – a reasoning, determining some property.
Theorem – a statement, determining some property and requiring a proof. Theorems are called also as lemmas, properties, consequences,
rules, criteria, propositions, statements. Proving a theorem, we are based on the earlier determined properties; some of them are also theorems. But some properties
are considered in geometry as main ones and are adopted without a proof.
Axiom – a statement, determining some property and adopted without a proof. Axioms have been arisen by experience and the experience checks if
they are true in totality. It is possible to build a set of axioms by different ways. But it is important that the adopted set of axioms would be sufficient to prove
all other geometrical properties and minimal. Changing one axiom in this set by another we must prove the replaced axiom, because now it is not an axiom,
but a theorem.
Initial notions. There are some notions in geometry ( and in mathematics in general ), to which it is impossible to give
some sensible definition. We adopt them as initial notions. The meaning of these notions can be ascertained only by experience. So, the notions
of a point and a straight line are initial. Basing on initial notions we can give definitions to all other notions.