are geometric items such as a point, a straight line, a circle, an angle, a triangle, etc.
Object Designations
Points
are designated via single upper case English alphabet characters:
$$A, B, C$$which can be distinguished further via indexing with natural numbers:
$$A_1, A_2, A_3$$or primes:
$$A', B', C'$$
Line Segments
are designated either via a two-point notation:
$$AB$$in which letters stand for points which mark the boundaries of the line segment
or via an arbitrary lower case English alphabet letter which may be further indexed with a natural number or primed:
$$AB = c$$which reads "assign a name \(c\) to a line segment between the points \(A\) and \(B\)".
Read Interpretation section about the rules of how exactly to interpret the meaning of the above assignment statement.
Straight Lines
are designated either via a functional notation:
$$Ln(A, B)$$which reads "a straight line passing through the points \(A\) and \(B\)"
or via an arbitrary lower case English alphabet letter which may be further indexed with a natural number or primed:
$$Ln(A, B) = p$$which reads "assign a name \(p\) to a straight line passing through the points \(A\) and \(B\)".
Line Segments
are designated either via a functional notation:
$$LnSeg(A, B)$$which reads "a segment of a straight line passing through the points \(A\) and \(B\)"
or via an arbitrary lower case English alphabet letter which may be further indexed with a natural number or primed:
$$LnSeg(A, B) = q$$which reads "assign a name \(q\) to a segment of a straight line passing through the points \(A\) and \(B\)".
Rays
are designated either via a functional notation:
$$Ray(A, B)$$which reads "a ray passing through the point \(B\) with an origin at the point \(A\)"
or via an arbitrary lower case English alphabet letter which may be further indexed with a natural number or primed:
$$Ray(A, B) = m$$which reads "assign a name \(m\) to a ray passing through the point \(B\) with origin at the point \(A\)".
Circles
are designated either via a functional notation:
$$Cir(O, r)$$which reads "a circle with a center at the point \(O\) and a radius \(r\)"
or via an arbitrary lower case English alphabet letter which may be further indexed with a natural number or primed:
$$Cir(O, OA) = q$$which reads "assign a name \(q\) to a circle with a center at the point \(O\) and a radius \(OA\)".
Though a circle is technically a plane figure, meaning a two-dimensional solid that encloses a certain amount of two-dimensional area, in the straight edge and compass constructions it is quite often taken to mean "a circumference". The context in which the word "circle" is used should make it unambiguously clear what exactly is meant.
Circular Arcs
are designated either via a functional notation:
$$Arc(O, AB)$$which reads "a circular arc between the points \(A\) and \(B\) of a circle with a center at the point \(O\)"
or via an arbitrary lower case English alphabet letter which may be further indexed with a natural number or primed:
$$Arc(O, AB) = q$$which reads "assign a name \(q\) to a circular arc between the points \(A\) and \(B\) of the circle with a center at the point \(O\)".
We remark that by definition either \(OA\) or \(OB\) in designations above are equal to the radius of the circle in question and that out of two possible arcs "the minor arc" is meant by default.
Angles
are designated either via a symbolic notation:
$$\angle ABC$$which reads "an angle with the vertex at the point \(B\) and legs on the straight lines \(BA\) and \(BC\)"
or via an arbitrary lower case Greek alphabet letter which may be indexed further with a natural number or primed:
$$\angle ABC = \alpha$$which reads "assign a name \(\alpha\) to \(\angle ABC\)".
We remark that normally with a symbolic angle designation the name of the vertex of an angle is kept in the middle:
$$\angle XYZ$$tells us that the vertex of this angle is located at the point \(Y\) and that its legs pass through the points \(X\) and \(Z\).
In rear cases when it is absolutely clear what is meant and no ambiguity can possibly arise it may be permissible to designate an angle via the name of a sole point - its vertex:
$$\angle A$$ $$\angle B$$ $$\angle C$$
Triangles
are designated symbolically via enumeration of their vertexes:
$$\triangle ABC$$for example, reads "a triangle with vertexes located at the points \(A\), \(B\) and \(C\)".
Other n-gons
are designated verbally:
- square \(ABCD\)
- rectangle \(ABCD\)
- pentagon \(ABCDE\)
- 5-gon \(ABCDE\)
and so on.
\(\blacksquare\)