SOLUTION: whats the difference between definition and theorem? example: definition of mid point vs mid point theorem?

Question 510131: whats the difference between definition and theorem?
example: definition of mid point vs mid point theorem?
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
Definitions do not need to be proven; they are simply definitions. A theorem is a statement that requires a proof, in which the proof is often based on previously proven theorems or axioms. It is important you know the difference between a definition and a theorem; when you have to prove something you need to make clear definitions and label your steps sequentially. Here are several examples of definitions/theorems from a wide variety of subjects:

Definition: for a given line segment there exists a midpoint.
Theorem: the midpoint divides the segments into two equal segments.
---------------------
Definition: we can define a complex number in the form a + bi where i is the imaginary unit.
Theorem:

(Euler's formula)
---------------------
Definition: given arbitrary non-negative real numbers

, ...,

and

, ...,

, we may assume

for

and

for

.
Theorem: The sum

is maximized with

and minimized with

(Rearrangement inequality).
---------------------
Definition: the altitudes of a triangle meet at a common point called the orthocenter.
Theorem: there exists such a point (this is a tricky one and the proof is not often covered in geometry classes).
---------------------
Definition: z is a root of the polynomial

if f(z) = 0.
Theorem: Any n-degree polynomial contains exactly n complex roots, including multiple roots (fundamental theorem of algebra).
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