a definition versus a theorem

Two kinds of true sentences sit at the heart of geometry, and beginners often blur them. One kind is true because we agreed to use a word that way; the other is true because we proved it. Telling a definition from a theorem is the difference between a rule of language and a discovery.

A definition assigns a precise meaning to a term — it is a naming convention, true by agreement and not requiring proof. 'An equilateral triangle is a triangle whose three sides are equal' is a definition: it simply tells you what the word means. A theorem, by contrast, is a statement that has been proved true from the axioms, definitions, and earlier theorems by a logical argument. It is a consequence we discover and must justify, not a meaning we declare. Supporting players include the postulate or axiom (an unproved starting assumption), the lemma (a small helper theorem), and the corollary (a result that follows quickly from a theorem).

Keeping the two apart matters because it shows you what still needs proving. The classic example lives in this very field: that an equilateral triangle has three equal sides is its definition (no proof needed), but that an equilateral triangle has three equal angles is a theorem (it must be proved, via the isosceles-triangle theorem). Likewise, the base angles of an isosceles triangle being equal is a theorem, not part of the definition. A common student error is to 'prove' something by quoting it as if it were the definition — but you cannot assume what you set out to prove.