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Hilbert's Undefined Terms and Definitions

    It wasn't until after the discovery of non-Euclidean geometry that mathematicians began examining the foundations of Euclidean geometry and formulating precise sets of axioms for it. The problem was to erect the entire structure of Euclidean geometry upon the simplest foundation possible; i.e. to choose a minimum number of undefined elements and relations and a set of axioms concerning them, with the property that all of the Euclidean geometry can be logically deduced form these without further appeal to intuition. Hilbert's approach does address Euclid's lack of attention to the notion of undefined terms and the concepts of incidence, betweenness and congruence. An example of Hilbert's precision and detail was to distinguish between a line and a line segment, as Euclid did not. This topic details Hilbert's undefined terms and preliminary definitions which can be used to provide the basis for traditional Euclidean geometry. A famous quote from Hilbert: "One must be able to say at all times-instead of points, lines, and planes---tables, chairs, and beer mugs."

    A mutual understanding of the following terms is assumed: point, line, lie on, between, congruent, set, element of, intersection, and union. For example, two lines intersect means there is one point that lies on both of them; or said differently, two lines are incident (have a point in common).

Definition (Line Segment) Given two points and , the line segment (or segment) is the set whose members are the points and , and all points that lie on the line and are between and . The two points and are called the endpoints of segment

Definition (Circle) Given two points and , the set of all points such that is called a circle with as the center and each of the segments is called a radius of the circle.

Definition (Ray) A ray is the following set of points: those points that belong to the segment and all points on such that is between and

Definition (Opposite Rays) Rays and are opposite rays if they are distinct, if they emanate from the same point and if they are part of the same line

Definition (Angle) An angle with vertex is a point together with two distinct non-opposite rays and (called the sides of the angle) emanating from

Definition (Supplementary Angles) If two angles and have a common side and the other two sides and form opposite rays, the angles are supplements of each other.

Definition (Right Angle) An angle is a right angle if it has a supplementary angle to which it is congruent.

Definition (Perpendicular) Two lines and are perpendicular if they intersect at a point and if there is a ray that is part of and a ray that is a part of such that is a right angle. Perpendicular lines and are denoted by

Definition (Midpoint) A point between and such that is called a midpoint of segment

Definition (Perpendicular Bisector) A perpendicular bisector is a line through a midpoint of a segment that is perpendicular to

Definition (Ray Bisects Angle) A ray bisects angle means

Definition (Collinear) Three points are collinear when they all lie on the same line.

Definition (Concurrent) Three lines are concurrent if they all pass through the same point.

Definition (Triangle) Given three non-collinear points and a triangle is the union of segments and The points   and are the vertices, the segments and are the sides, and the angles and are the angles of triangle Given one of the three vertices, the opposite side to the vertex is the segment not containing it, the adjacent sides to the vertex are the other two.

Definition (Medians of a Triangle) A median of a triangle is a segment from a vertex to a midpoint of its opposite side.

Definition (Altitudes of a Triangle) An altitude of a triangle is a segment such that is a vertex, lies on the line containing the opposite side to and

Definition (Isosceles Triangle) A triangle with at least two sides congruent is called an isosceles triangle.

Definition (Equilateral Triangle) A triangle with all three sides congruent is called an equilateral triangle.

Definition (Right Triangle) A triangle with a right angle is called a right triangle.

Definition (Parallel Lines) Two lines are parallel if no point lies on both of them.

Definition (Quadrilateral) Given four points and no three of which are collinear and such that any pair of the segments and either have no point in common or have only an endpoint in common, then a quadrilateral (denoted by ) is the union of the segments and The points and are the vertices of the quadrilateral and the segments and are the sides of the quadrilateral.

Definition (Adjacent and Opposite Sides of a Quadrilateral) Two sides of a quadrilateral that have a common vertex are called adjacent sides of a quadrilateral and two sides that do not have a common vertex are called opposite sides of a quadrilateral.

Definition (Diagonals of a Quadrilateral) The diagonals of a quadrilateral are the segments that are not sides of the quadrilateral whose endpoints are vertices of the quadrilateral.

Definition (Parallelogram) A parallelogram is a quadrilateral such that for each pair of opposite sides, the lines containing those sides are parallel.

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