# Definitions and concepts

## Concept of postulate, theorem, hypothesis, thesis and corollary

On this page you can find the definition and concept of some mathematical and geometric terms such as proposition, Axiom, Theorem, Postulate, Lemma, Corollary, Scolium and Problem among others. Find the concept of postulate, theorem, hypothesis, thesis and corollary

## Proposition

The concept proposition is a statement of a hypothesis or assumption, and of a thesis or conclusion, which is a consequence of the hypothesis.

Example :
Proposition 3 of Book III of the Elements of Euclid: If in a circle a line CD drawn through the center E divides in two equal parts another line AB not drawn through the center, it cuts it also forming right angles; and if you cut it at right angles, you also divide it into two equal parts AF and FB.

## Axiom

The concept “Axiom” is a self-evident proposition and therefore needs no proof.
For examples we have the Euclidean axioms: The whole is equal to the sum of the parts. The whole is greater than each of the parts.

Between two points a single straight line passes.

## Theorem

The Theorem as a concept is a proposition that needs proof to be evident. For example: The sum of the angles of a triangle is equal to two right angles.Example: If two parallel lines intersect with a secant line, the following angle relation is fulfilled:

• Alternate / interior angles are the same.
• The alternate / exterior angles are the same.
• The corresponding angles are the same.
• Internal collateral angles are supplemental.
• External collateral angles are supplemental.

## Dual theorem

The principle of duality affirms that from any theorem or construction of projective geometry we can obtain another, known as the dual theorem, only the words point and line can be exchanged, also modifying the relationships between the points and the lines. So by this principle,

• A point becomes a line.
• Aligned points become lines that pass through a point.
• Tangent lines become the point of tangency.
• A circumscribed circle becomes an inscribed circle.
• etc.

The dual theorem of Pascal’s theorem is Brianchon’s theorem.

## Pascal’s theorem

Any hexagon inscribed in a circle, the intersection points of the opposite sides are in a straight line.

Pascal’s theorem, discovered by Blaise Pascal (1623-1662) at the age of 16, refers to aligned points: If the 6 vertices of a hexagon are located on a conic and the three pairs of opposite sides intersect, then the intersection points are aligned. The line containing the three points of intersection is known as Pascal’s line

Here you can find a more detailed explanation of Pascal’s theorem.

## Brianchon’s theorem

Brianchon’s theorem is due to Charles Julien Brianchon (1783-1864) and states that the diagonals of a hexagon circumscribed to a conic intersect at one point.

Here you can find a more detailed explanation of Brianchon’s theorem.

## Feuerbach’s theorem

The Euler or 9-point circle is tangent to the inscribed and ex-inscribed circles to the triangle.

## Theorem and Gauss

The midpoints of the diagonals of an entire quadrilateral are in a straight line.

## Euler’s theorem

In any convex polyhedron, the number of faces plus the number of vertices is equal to the number of edges plus two. (faces + vertices = edges + 2).

## Postulate

Postulate is a proposition that is admitted without proof, although without the evidence of the axiom. For example: For a point outside a line, you can only draw a single parallel to the line.

## Lemma

Lemma is a preliminary theorem that serves as a basis to prove other propositions.

## Corollary

Corollary or consequence is a theorem the truth of which is simply deduced from another already proven.

## Scholium

Escolio is a warning or note that is made in order to clarify, extend or restrict previous propositions.

## Problem

Problem is a question that is proposed with the purpose and intention of clarifying or solving it using a certain methodology.

`Agradecimiento por la colaboración deFrancisco Javier García Capitán, 2000`
`Copyright Applet © 1996/1997 (Juny, 1997) D.E.Joyce Clark University© Drets d´ús cedits 2002/2003JDLPrimera Edición Castellano-Catalán ®2006`