Geometry :: Notes :: Neutral Geometry
1 Neutral Geometry
The incidence geometry given in Incidence Geometry had only three axioms and as a result has limited complexity. For example, the Euclidean Plane most people are familiar with from high school has a much more complex structure which involves distances and angles. In order to work with more complex structures, such as the Euclidean Plane, we will need a more robust set of axioms. The axioms in this section will give rise to the neutral geometry, and theorems proved in this section will not only be valid for Euclidean Geometry, but also for Elliptical (Spherical) Geometry and Hyperbolic Geometry. Similar to what was done in incidence geometry with Elliptic Axiom, Euclidean Axiom, and ./axiom-incidence-axiom-hyperbolic.html, we will add additional axioms for the above three geometries. It should be noted however that much more can be done with Incidence Geometry than we have, see https://en.wikipedia.org/wiki/Incidence_geometry} for example.
The undefined terms for the neutral geometry are: point, line, distance, half-plane,angle measure, and area.
1.1 Existence Axiom
The collection of all points forms a nonempty set, and there is more than one point in that set.
The set of all points is called the plane and is denoted by \(\mathbb{P}\).
1.2 Incidence Axiom
Every line is a set of points. For every pair of distinct points \(A\) and \(B\) there is exactly one line \(l\) such that \(A \in l\) and \(B \in l\).
For distinct points \(A\) and \(B\), we use the notation \(\overleftrightarrow{AB}\) to denote the line determined by \(A\) and \(B\).
A point \(P\) is said to lie on line \(l\) if \(P \in l\).
A point \(P\) is called an external point for line \(l\) if \(P\) does not lie on \(l\).
Lines \(l\) and \(m\) are parallel if they are distinct and no point is incident with both of them, ie \(l \cap m = \emptyset\). If \(l\) and \(m\) are parallel we write \(l \parallel m\).
For lines \(l\) and \(m\), either \(l=m\), \(l\parallel m\), or \(l \cap m = \{x\}\) for some point \(x \in \mathbb{P}\).
1.3 Distance Axiom
For every pair of points \(P\) and \(Q\) there exists a real number \(PQ\), called the distance from \(P\) to \(Q\). For each line \(l\) there is a one-to-one correspondence from \(l\) to \(\mathbb{R}\) such that if \(P\) and \(Q\) are points on the line that correspond to the real numbers \(x\) and \(y\), respectively, then \(PQ = |x-y|\).
Three or more points \(A,B,C, \ldots\) are collinear if there exists a line incident with all of them. The points are noncollinear if no such line exists.
For distinct points \(A,B,\) and \(C\), we say \(C\) is between \(A\), and \(B\) and write \(A\cdot C\cdot B\), if \(A,B,\) and \(C\) are collinear, and \(AC+CB = AB\).
The segment between two points \(A\) and \(B\) is
\begin{equation*} \overline{AB} = \{A,B\} \cup \{P | A \cdot P \cdot B\}. \end{equation*}The points \(A\) and \(B\) are called the end points of the segment \(\overline{AB}\). All other points of \(\overline{AB}\) are called interior points.
The ray between two distinct points \(A\) and \(B\) is
\begin{equation*} \overrightarrow{AB} = \overline{AB} \cup \{P | A \cdot B \cdot P\}. \end{equation*}The point \(A\) is the endpoint of the ray \(\overrightarrow{AB}\).
The length of a segment \(\overline{AB}\) is the distance from \(A\) to \(B\) and is denoted \(AB\).
Two segments \(\overline{AB}\) and \(\overline{CD}\) are said to be congruent if they have the same length, we write \(\overline{AB} \cong \overline{CD}\).
A metric is a function \(d:\mathbb{P}^2 \to \mathbb{R}\) where for all \(P,Q,R \in \mathbb{P}\)
- \(d(P,Q) \geq 0\), with equality if and only if \(P = Q\);
- \(d(P,Q) = d(Q,P)\);
- \(d(P,Q) \leq d(P,R) + d(R,Q)\).
As a function, length is a metric.
\(A\cdot C \cdot B\) if and only if \(B \cdot C \cdot A\).
The Euclidean metric is defined by
\[ d((x_0,y_0),(x_1,y_1)) = \sqrt{(x_1 - x_0)^2 + (y_1 - y_0)^2}, \]
where \((x_0,y_0), (x_1,y_1) \in \mathbb{P}\). Show that the Euclidean metric is in fact a metric. Draw a unit circle in the Euclidean metric. A unit circle is \(\{(x,y) \in \mathbb{P} \,|\, d((0,0),(x,y)) = 1\}\). Hint: For the triangle inequality, see the linear algebra review.
The taxicab metric is defined by
\[ \rho((x_0,y_0),(x_1,y_1)) = |x_1 - x_0| + |y_1 - y_0|, \]
where \((x_0,y_0), (x_1,y_1) \in \mathbb{P}\). Show that the taxicab metric is in fact a metric. Draw a unit circle in the taxicab metric. A unit circle is \(\{(x,y) \in \mathbb{P} \,|\, \rho((0,0),(x,y)) = 1\}\). Hint: See Theorem Length is a Metric.
Investigate the taxicab metric in more detail. For example, what do equilateral triangles look like? What do ellipses look like? Why is it called the taxicab metric? For more information and project ideas see Section 3.2 of the textbook.
Let \(A, B \in \mathbb{S}^2\). Define the spherical distance \(s(A,B)\) to be the length of the shortest arc of a great circle containing \(A\) and \(B\). Show that \(s\) is in fact a metric. What is the greatest distance between two points?
A coordinate function for a line \(l\) is a one-to-one and onto function \(f:l \to \mathbb{R}\) so that for every \(P,Q \in l\), \(PQ = |f(P) - f(Q)|\). The number \(f(P) \in \mathbb{R}\) is called the coordinate of the point \(P\).
Let \(l\) be a line in the Cartesian plane. If \(l\) is not a vertical line, then \(l\) has an equation of the form \(y=mx+b\). In that case define \(f:l \to \mathbb{R}\) by \(f(x,y) = x \sqrt{1+m^2}\). If \(l\) is a vertical line, then it has an equation of the form \(x=a\). In this case define \(f:l \to \mathbb{R}\) by \(f(a,y) = y\). Show that \(f\) is a coordinate function in the Euclidean metric.
Let \(l\) be a line in the Cartesian plane. Define \(f:l \to \mathbb{R}\) by
\begin{equation*} f(x,y) = \begin{cases} x(1 + |m|) & \textrm{ if \(l\) is not a vertical line} \\ y & \textrm{ if \(l\) is a vertical line}. \end{cases} \end{equation*}Show that \(f\) is a coordinate function in the taxicab metric. See Example Coordinate Functions in the Euclidean Metric for definitions of \(m\) and \(a\).
For every pair of distinct points \(P\) and \(Q\), there is a coordinate function \(f: \overleftrightarrow{PQ} \to \mathbb{R}\) so that \(f(P) = 0\) and \(0 < f(Q)\).
Let \(l\) be a line and \(A,B\), and \(C\) three distinct points on \(l\). Let \(f:l \to \mathbb{R}\) be a coordinate function for \(l\). The point \(C\) is between \(A\) and \(B\) if and only if either \(f(A) < f(C) < f(B)\) or \(f(A) > f(C) > f(B)\).
Let \(A,B,\) and \(C\) be three points such that \(B\) lies on \(\overrightarrow{AC}\). Then \(A \cdot B \cdot C\) if and only if \(AB < AC\).
If \(A,B,\) and \(C\) are three distinct collinear points, then exactly one of them lies between the other two.
Let \(A\) and \(B\) be two distinct points. If \(f\) is a coordinate function for line \(l = \overleftrightarrow{AB}\) such that \(f(A) = 0\) and \(f(B) >0\), then \(\overrightarrow{AB} = \{P \in l\, |\, f(P) \geq 0\}\).
Let \(A\) and \(B\) be two distinct points. The point \(M\) is called a midpoint of \(\overline{AB}\) if \(M\) is between \(A\) and \(B\) and \(AM=MB\).
If \(A\) and \(B\) are distinct points and \(d\) is any nonnegative real number, then there exists a unique point \(C\) such that \(C\) lies on \(\overrightarrow{AB}\) and \(AC=d\).
1.4 Plane Separation Axiom
A set of points \(S\) is convex if for every pair of points \(A,B\in S\), the segment \(\overline{AB} \subset S\).
For every line \(l\), the points that do not lie on \(l\) form two disjoint, nonempty sets \(H_1\) and \(H_2\), called half-planed bounded by \(l\), such that:
- each of \(H_1\) and \(H_2\) is convex;
- if \(p \in H_1\) and \(Q \in H_2\), then \(\overline {PQ}\) intersects \(l\).
Let \(l\) be a line and let \(A\) be an external point. We use \(H_A\) to denote the half-plane bounded by \(l\) which contains \(A\).
Let \(l\) be a line and \(H_1,H_2\) the two half-planes bounded by \(l\). Let \(A\) and \(B\) be external points. \(A\) and \(B\) are said to be on the same side of \(l\) if they are both in \(H_1\) or both in \(H_2\). Otherwise they are said to be on opposite sides of \(l\).
Let \(l\) be a line with \(A\) and \(B\) points not on \(l\). The points \(A\) and \(B\) are on the same side of \(l\) if and only if \(\overline{AB} \cap l = \emptyset\). The points are on opposite sides of \(l\) if and only if \(\overline{AB} \cap l \neq \emptyset\).
Two rays \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\) which have the same end point are opposite rays if they are not equal, but \(\overleftrightarrow{AB} = \overleftrightarrow{AC}\). Otherwise they are non-opposite.
An angle is the union of two non-opposite rays \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\). The angle is denoted by \(\angle BAC\) or \(\angle CAB\). The point \(A\) is called the vertex, and the rays \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\) are called the sides of the angle.
Let \(A,B\) and \(C\) be points so that \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\) are non-opposite rays. The interior of angle \(\angle BAC\) is
- \(\emptyset\) if \(\overrightarrow{AB} = \overrightarrow{AC}\)
- \(H_B \cap H_C\) if \(\overrightarrow{AB} \neq \overrightarrow{AC}\), where \(H_B\) is the half-plane determined by \(\overleftrightarrow{AC}\) and \(B\), and \(H_C\) is the half-plane determined by \(\overleftrightarrow{AB}\) and \(C\).
Ray \(\overrightarrow{AD}\) is between rays \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\) if \(D\) is in the interior of \(\angle BAC\).
Let \(l\) be a line, \(A \in l\), and \(B\) an external point of \(l\). If \(C\) is a point on \(\overrightarrow{AB}\) and \(C\neq A\), then \(B\) and \(C\) are on the same side of \(l\).
Let \(A,B\), and \(C\) be three noncollinear points and let \(D\) be a point on the line \(\overleftrightarrow{BC}\). The point \(D\) is between points \(B\) and \(C\) if and only if ray \(\overrightarrow{AD}\) is between rays \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\).
Let \(A,B\), and \(C\) be noncollinear points. The triangle ABC is
\begin{equation*} \triangle ABC = \overline{AB} \cup \overline{BC} \cup \overline{AC}. \end{equation*}The points \(A,B\) and \(C\) are the vertices of the triangle, while the segments \(\overline{AB}, \overline{BC}\) and \(\overline{AC}\) are its sides.
Let \(\triangle ABC\) be a triangle and let \(l\) be a line such that none of \(A,B\) or \(C\) lies on \(l\). If \(l\) intersects \(\overline{AB}\), then \(l\) intersects either \(\overline{AC}\) or \(\overline{BC}\).
1.5 Protractor Axiom
For every angle \(\angle BAC\) there is a real number \(\mu(\angle BAC)\), called the measure of \(\angle BAC\), such that:
- \(0^\circ \leq \mu(\angle BAC) < 180^\circ\) for every angle \(\angle BAC\);
- \(\mu(\angle BAC) = 0^\circ\) if and only if \(\overrightarrow{AB} = \overrightarrow{AC}\).
- For each real number \(r \in (0,180)\), and for each half-plane \(H\) bounded by \(\overleftrightarrow{AB}\) there exists a unique ray \(\overrightarrow{AE}\) such that \(E\) is in \(H\) and \(\mu(\angle BAE) = r^\circ\).
- If the ray \(\overrightarrow{AD}\) is between rays \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\), then \(\mu(\angle BAD) + \mu(\angle DAC) = \mu(\angle BAC)\).
Two angles \(\angle BAC\) and \(\angle EDF\) are congruent if \(\mu(\angle BAC) = \mu(\angle EDF)\). Angle congruence is denoted by \(\angle BAC \cong \angle EDF\).
Angle \(\angle BAC\) is a(n):
- acute angle if \(\mu(\angle BAC) < 90^\circ\),
- right angle if \(\mu(\angle BAC) = 90^\circ\),
- obtuse angle if \(\mu(\angle BAC) > 90^\circ\).
If \(A,B,C\), and \(D\) are four distinct points such that \(C\) and \(D\) are on the same side of \(\overleftrightarrow{AB}\) and \(D\) is not on \(\overrightarrow{AC}\), then either \(C\) is in the interior of \(\angle BAD\) or \(D\) is in the interior of \(\angle BAC\).
Let \(A,B,C\), and \(D\) be four distinct points such that \(C\) and \(D\) lie on the same side of \(\overleftrightarrow{AB}\). Then \(\mu(\angle BAD) < \mu(\angle BAC)\) if and only if \(\overrightarrow{AD}\) is between rays \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\).
Let \(A,B\), and \(C\) be three noncollinear points. A ray \(\overrightarrow{AD}\) is an angle bisector of \(\angle BAC\) if \(D\) is in the interior of \(\angle BAC\) and \(\mu(\angle BAD) = \mu(\angle DAC)\).
Let \(A\) and \(B\) be two points, \(H\) be one of the half-planes of \(l = \overleftrightarrow{AB}\), and let \(\mathcal{A} = \{\angle BAE \, | \, E \in H\}\). Define \(f:\mathcal{A} \to (0,180)\) by \(f(\angle BAE) = \mu(\angle BAE)\). Then \(f\) is a one-to-one and onto function, and \(\overrightarrow{AF}\) is between \(\overrightarrow{AB}\) and \(\overrightarrow{AE}\) if and only if \(f(\angle BAF)\) is between 0 and \(f(\angle BAE)\).
If \(A,B\), and \(C\) are three noncollinear points, then there exists a unique angle bisector for \(\angle BAC\).
Let \(l\) be a line and \(A\) and \(D\) distinct points on \(l\). If \(B\) and \(E\) are points on opposite sides of \(l\), then \(\overrightarrow{AB} \cap \overrightarrow{DE} = \emptyset\).
If \(\triangle ABC\) is a triangle and \(D\) is a point in the interior of \(\angle BAC\), then there is a point \(G\) such that \(G\) lies on both \(\overrightarrow{AD}\) and \(\overline{BC}\).
A point \(D\) is in the interior of the angle \(\angle BAC\) if and only if \(\overrightarrow{AD}\) intersects the interior of \(\overline{BC}\).
Two angles \(\angle BAD\) and \(\angle DAC\) form a linear pair if \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\) are opposite rays.
Let \(A,B,C,D\), and \(E\) be points. If \(C\cdot A \cdot B\) and \(D\) is in the interior of \(\angle BAE\), then \(E\) is in the interior of \(\angle DAC\).
If angles \(\angle BAD\) and \(\angle DAC\) form a linear pair, then \(\mu(\angle BAD) + \mu(\angle DAC) = 180^\circ\).
Two lines \(l\) and \(m\) are perpendicular if there exists a point \(A\) that lies on both \(l\) and \(m\) and there are points \(B \in l\) and \(C \in m\) so that \(\angle BAC\) is a right angle. This is denoted \(l \perp m\).
If \(l\) is a line and \(P\) is a point on \(l\), then there exists exactly one line \(m\) such that \(P\) lies on \(m\) and \(m \perp l\).
Let \(D\) and \(E\) be distinct points. A perpendicular bisector of \(\overline{DE}\) is a line \(l\) such that the midpoint of \(\overline{DE}\) lies on \(l\) and \(l \perp \overleftrightarrow{DE}\).
If \(D\) and \(E\) are two distinct points, then there exists a unique perpendicular bisector for \(\overline{DE}\).
Angles \(\angle BAC\) and \(\angle DAE\) are called vertical angles if rays \(\overrightarrow{AB}\) and \(\overrightarrow{AE}\) are opposite and rays \(\overrightarrow{AC}\) and \(\overrightarrow{AD}\) are opposite, or if rays \(\overrightarrow{AB}\) and \(\overrightarrow{AD}\) are opposite and \(\overrightarrow{AC}\) and \(\overrightarrow{AE}\) are opposite.
Vertical angles are congruent.
Two triangles are congruent if there is a correspondence between the vertices of the first triangle and the vertices of the second triangle such that corresponding angles are congruent and corresponding sides are congruent.
If \(\triangle ABC\) and \(\triangle DEF\) are congruent, we write \(\triangle ABC \cong \triangle DEF\). We take this to mean that \(A \leftrightarrow D\), \(B \leftrightarrow E\) \(C \leftrightarrow F\); thus \(\overline{AB} \cong \overline{DE}\), \(\overline{BC} \cong \overline{EF}\), \(\overline{AC} \cong \overline{DF}\), \(\angle{ABC} \cong \angle{DEF}\) \(\angle{BCA} \cong \angle{EDF}\), and \(\angle{CAB} \cong \angle{FDE}\).
1.6 Side Angle Side Axiom
If \(\triangle ABC\) and \(\triangle DEF\) are two triangles such that \(\overline{AB} \cong \overline{DE}\), \(\angle ABC \cong \angle DEF\) and \(\overline{BC} \cong \overline{EF}\), then \(\triangle ABC \cong \triangle DEF\).
A triangle is an isosceles triangle if it has a pair of congruent sides. The two angles not between the congruent sides are called base angles.
The base angles of an isosceles triangle are congruent.
The Side angle Side axiom does not hold in the taxicab geometry. As a result, the taxicab metric is not a model for the neutral geometry. (It does satisfy the other five axioms but you need not show this).