hcju/proofwikips huggingface dataset:
Definition:Apotome/Fifth Apotome
Let $a, b \in \set {x \in \R_{>0} : x^2 \in \Q}$ be two rationally expressible numbers such that $a - b$ is an apotome. Then $a - b$ is a '''fifth apotome''' : :$(1): \quad b \in \Q$ :$(2): \quad \dfrac {\sqrt {a^2 - b^2}} a \notin \Q$ where $\Q$ denotes the set of rational numbers.  :''{{Definition:Euclid's Definitions - Book X (III)/5 - Fifth Apotome}}'' {{EuclidDefRefNocat|X (III)|5|Fifth Apotome}}
Definition:Apotome/Sixth Apotome
Let $a, b \in \set {x \in \R_{>0} : x^2 \in \Q}$ be two rationally expressible numbers such that $a - b$ is an apotome. Then $a - b$ is a '''sixth apotome''' : :$(1): \quad a \notin \Q$ :$(2): \quad b \notin \Q$ :$(3): \quad \dfrac {\sqrt {a^2 - b^2}} a \notin \Q$ where $\Q$ denotes the set of rational numbers.  :''{{Definition:Euclid's Definitions - Book X (III)/6 - Sixth Apotome}}'' {{EuclidDefRefNocat|X (III)|6|Sixth Apotome}}
Definition:Line/Straight Line Segment
A '''straight line segment''' is a line segment which is straight.  :''A straight line segment can be drawn joining any two points.'' ({{BookLink|The Elements|Euclid}}: Postulates: Euclid's Second Postulate)
Definition:Line/Segment
A '''line segment''' is any line (straight or not) which terminates at two points. === Straight Line Segment === {{:Definition:Line/Straight Line Segment}} === Endpoint === {{:Definition:Line/Endpoint}} === Midpoint === {{:Definition:Line/Midpoint}}
Definition:Line/Infinite Half-Line
An '''infinite half-line''' is a line which terminates at an endpoint at one end, but has no such endpoint at the other.
Definition:Line/Equality
Two line segments are '''equal'''  they have the same length.
Definition:Incircle
Let $P$ be a polygon. Let $C$ be a circle which is inscribed within $P$. :300px Then $C$ is an '''incircle''' of $P$. === Incenter === {{:Definition:Incircle/Incenter}} === Inradius === {{:Definition:Incircle/Inradius}}
Definition:Circumcircle
Let $P$ be a polygon. Let $C$ be a circle which is circumscribed around $P$. :300px Then $C$ is a '''circumcircle''' of $P$.
Definition:Incircle of Triangle/Incenter
The center of an incircle of a triangle is called an '''incenter of the triangle'''.
Definition:Incircle of Triangle/Inradius
A radius of an incircle of a triangle is called an '''inradius of the triangle'''. In the above diagram, $r$ is an inradius.
Definition:Circumcircle of Triangle/Circumcenter
The center of a circumcircle of a triangle is called a '''circumcenter of a triangle'''.
Definition:Circumcircle of Triangle/Circumradius
A radius of a circumcircle of a triangle is called a '''circumradius of a triangle'''. In the above diagram, $R$ is a circumradius.
Definition:Circumcenter
Let $P$ be a polygon. Let $P$ have a circumcircle $C$. :300px Then the center of $C$ is the '''circumcenter''' of $P$.
Definition:Excircle of Triangle/Exradius
A radius of an excircle of a triangle is called an '''exradius of the triangle'''. In the above diagram, $r$ is an exradius.
Definition:Excircle of Triangle/Excenter
The center of an excircle of a triangle is called an '''excenter of the triangle'''.
Definition:Tangent/Geometry/Tangent Line
 :''{{:Definition:Euclid's Definitions - Book III/2 - Tangent to Circle}}'' {{EuclidDefRefNocat|III|2|Tangent to Circle}} :300px In the above diagram, the line is '''tangent''' to the circle at the point $C$.
Definition:Tangent/Geometry/Tangent Circles
 :''{{:Definition:Euclid's Definitions - Book III/3 - Tangent Circles}}'' {{EuclidDefRefNocat|III|3|Tangent Circles}} :400px In the above diagram, the two circles are '''tangent''' to each other at the point $C$.
Definition:Piecewise Continuous Function
Let $f$ be a real function defined on a closed interval $\left[{a \,.\,.\, b}\right]$. $f$ is '''piecewise continuous''' : :there exists a finite subdivision $\left\{ {x_0, x_1, \ldots, x_n}\right\}$ of $\left[{a \,.\,.\, b}\right]$, where $x_0 = a$ and $x_n = b$, such that: ::for all $i \in \left\{ {1, 2, \ldots, n}\right\}$, $f$ is continuous on $\left({x_{i − 1} \,.\,.\, x_i}\right)$.
Definition:Piecewise Continuously Differentiable Function
Let $f$ be a real function defined on a closed interval $\closedint a b$. === Definition 1 === {{:Definition:Piecewise Continuously Differentiable Function/Definition 1}} === Definition 2 === {{:Definition:Piecewise Continuously Differentiable Function/Definition 2}}
Definition:Cone/Double Napped Cone
A '''double napped cone''' is a cone where the lines joining the apex to the circumference of the base extend indefinitely in either dimension: :300px
Definition:Cone/Double Napped Cone/Nappe
Each of the halves of a double napped cone is a '''nappe'''.
Definition:Plane Surface
{{EuclidDefinition|book = I|def = 7|name = Plane Surface}}
Definition:Plane Surface/The Plane
'''The plane''' is the term used for the general plane surface which is infinite in all directions.
Definition:Cone/Apex
In the above diagram, the point $A$ is known as the '''apex''' of the cone.
Definition:Cone/Base
The plane figure $PQR$ is called the '''base''' of the cone.
Definition:Right Circular Cone/Opening Angle
Let $K$ be a right circular cone. Let point $A$ be the apex of $K$. Let $B$ and $C$ be the endpoints of a diameter of the base of $K$. Then the angle $\angle BAC$ is the '''opening angle''' of $K$. In the above diagram, $\phi$ is the '''opening angle''' of the right circular cone depicted.
Definition:Conic Section
=== Intersection with Cone === {{:Definition:Conic Section/Intersection with Cone}} === Focus-Directrix Property === {{:Definition:Conic Section/Focus-Directrix Property}}
Definition:Ellipse
=== Intersection with Cone === {{:Definition:Conic Section/Intersection with Cone}} {{:Definition:Conic Section/Intersection with Cone/Ellipse}} === Focus-Directrix Property === {{:Definition:Ellipse/Focus-Directrix}} === Equidistance Property === {{:Definition:Ellipse/Equidistance}}
Definition:Parallel (Geometry)/Lines
 :''{{:Definition:Euclid's Definitions - Book I/23 - Parallel Lines}}'' {{EuclidDefRefNocat|I|23|Parallel Lines}} The contemporary interpretation of the concept of parallelism declares that a straight line is parallel to itself.
Definition:Parallel (Geometry)/Planes
Two planes are '''parallel''' , when produced indefinitely, do not intersect at any point. {{EuclidDefinition|book = XI|def = 8|name = Parallel Planes}} The contemporary interpretation of the concept of parallelism declares that a plane is parallel to itself.
Definition:Right Circular Cone/Directrix
Let $K$ be a right circular cone. Let $B$ be the base of $K$. The circumference of $B$ is the '''directrix''' of $K$.
Definition:Right Circular Cone/Generatrix
Let $K$ be a right circular cone. Let $A$ be the apex of $K$. Let $B$ be the base of $K$. Then a line joining the apex of $K$ to its directrix is a '''generatrix of $K$'''.
Definition:Generatrix
A '''generatrix''' is an element of a set of straight lines which constitute a given surface.
Definition:Conic Section/Intersection with Cone/Parabola
:400px Let $\phi = \theta$. Then $K$ is a parabola.
Definition:Conic Section/Intersection with Cone/Circle
:400px Let $\phi = \dfrac \pi 2 - \theta$, thereby making $D$ perpendicular to the axis of $C$. Then $D$ and $B$ are parallel, and so $K$ is a circle.
Definition:Conic Section/Intersection with Cone/Ellipse
:400px Let $\theta < \phi < \dfrac \pi 2 - \theta$. That is, the angle between $D$ and the axis of $C$ is between that for which $K$ is a circle and that which $K$ is a parabola. Then $K$ is an ellipse.
Definition:Conic Section/Intersection with Cone/Hyperbola
:400px Let $\phi < \theta$. Then $K$ is a hyperbola. Note that in this case $D$ intersects $C$ in two places: one for each nappe of $C$.
Definition:Ellipse/Focus-Directrix
400px Let $D$ be a straight line. Let $F$ be a point. Let $\epsilon \in \R: 0 < \epsilon < 1$. Let $K$ be the locus of points $b$ such that the distance $p$ from $P$ to $D$ and the distance $q$ from $P$ to $F$ are related by the condition: :$\epsilon \, p = q$ Then $K$ is an '''ellipse'''.
Definition:Ellipse/Equidistance
:400px Let $F_1$ and $F_2$ be two points in the plane. Let $d$ be a length greater than the distance between $F_1$ and $F_2$. Let $K$ be the locus of points $P$ which are subject to the condition: :$d_1 + d_2 = d$ where: :$d_1$ is the distance from $P$ to $F_1$ :$d_2$ is the distance from $P$ to $F_2$. Then $K$ is an '''ellipse'''. This property is known as the '''equidistance property'''.
Definition:Line/Midpoint
Let $L = AB$ be a line segment whose endpoints are $A$ and $B$. Let $M$ be a point on $L$ such that the line segment $AM$ is equal to the line segment $MB$. Then $M$ is the '''midpoint''' of $L$.
Definition:Ellipse/Major Axis
The '''major axis''' of $K$ is the line segment passing through both $F_1$ and $F_2$ whose endpoints are where it intersects $K$.
Definition:Ellipse/Minor Axis
The '''minor axis''' of $K$ is the line segment through the center of $K$ perpendicular to the major axis of $K$ such that its endpoints are the points of intersection with $K$.
Definition:Ellipse/Minor Axis/Semi-Minor Axis
A '''semi-minor axis''' of $K$ is either half of the minor axis of $K$ from its midpoint to its endpoint.
Definition:Ellipse/Major Axis/Semi-Major Axis
A '''semi-major axis''' of $K$ is either half of the major axis of $K$ from its midpoint to its endpoint.
Definition:Ellipse/Covertex
A '''covertex''' of $K$ is either one of the endpoints of the minor axis of $K$.
Definition:Ellipse/Vertex
A '''vertex''' of $K$ is either one of the endpoints of the major axis of $K$.
Definition:Cartesian Plane
324pxrightthumbA general point $Q = \tuple {x, y}$ in the Cartesian plane The '''Cartesian plane''' is a Cartesian coordinate system of $2$ dimensions. Every point on the plane can be identified uniquely by means of an ordered pair of real coordinates $\tuple {x, y}$, as follows: Identify one distinct point on the plane as the origin $O$. Select a point $P$ on the plane different from $O$. Construct an infinite straight line through $O$ and $P$ and call it the $x$-axis. Identify the $x$-axis with the real number line such that: :$0$ is identified with the origin $O$ :$1$ is identified with the point $P$ The orientation of the $x$-axis is determined by the relative positions of $O$ and $P$. It is conventional to locate $P$ to the right of $O$, so as to arrange that: :to the right of the origin, the numbers on the $x$-axis are positive :to the left of the origin, the numbers on the $x$-axis are negative. Construct an infinite straight line through $O$ perpendicular to the $x$-axis and call it the $y$-axis. Identify the point $P'$ on the $y$-axis such that $OP' = OP$. Identify the $y$-axis with the real number line such that: :$0$ is identified with the origin $O$ :$1$ is identified with the point $P'$ The orientation of the $y$-axis is determined by the position of $P'$ relative to $O$. It is conventional to locate $P'$ such that, if one were to imagine being positioned at $O$ and facing along the $x$-axis towards $P$, then $P'$ is on the left.  Hence with the conventional orientation of the $x$-axis as horizontal and increasing to the right: :going vertically "up" the page or screen from the origin, the numbers on the $y$-axis are positive :going vertically "down" the page or screen from the origin, the numbers on the $y$-axis are negative.
Definition:Cartesian Coordinate System/X Coordinate
Let $x$ be the length of the line segment from the origin $O$ to the foot of the perpendicular from $Q$ to the $x$-axis. Then $x$ is known as the '''$x$ coordinate'''. If $Q$ is in the positive direction along the real number line that is the $x$-axis, then $x$ is positive. If $Q$ is in the negative direction along the real number line that is the $x$-axis, then $x$ is negative.
Definition:Cartesian Coordinate System/Y Coordinate
Let $y$ be the length of the line segment from the origin $O$ to the foot of the perpendicular from $Q$ to the $y$-axis. Then $y$ is known as the '''$y$ coordinate'''. If $Q$ is in the positive direction along the real number line that is the $y$-axis, then $y$ is positive. If $Q$ is in the negative direction along the real number line that is the $y$-axis, then $y$ is negative.
Definition:Cartesian Plane/Quadrants
For ease of reference, the cartesian plane is often divided into four quadrants by the axes: === First Quadrant === {{:Definition:First Quadrant}} === Second Quadrant === {{:Definition:Second Quadrant}} === Third Quadrant === {{:Definition:Third Quadrant}} === Fourth Quadrant === {{:Definition:Fourth Quadrant}} Note that the axes themselves are generally not considered to belong to any quadrant.
Definition:Axis/Coordinate Axes
Consider a coordinate system. One of the reference lines of such a system is called an '''axis'''.
Definition:Axis/X-Axis
In a cartesian coordinate system, the '''$x$-axis''' is the one usually depicted and visualised as going from left to right. It consists of all the points in the real vector space in question (usually either $\R^2$ or $\R^3$) at which all the elements of its coordinates but $x$ are zero.
Definition:Axis/Y-Axis
In a cartesian coordinate system, the '''$y$-axis''' is the one usually depicted and visualised as going from "bottom" to "top" of the paper (or screen). It consists of all the points in the real vector space in question (usually either $\R^2$ or $\R^3$) at which all the elements of its coordinates but $y$ are zero.
Definition:Axis/Z-Axis/Right-Hand Rule
The usual convention for the orientation of the $z$-axis is that of the right-hand rule: Let the coordinate axes be oriented as follows: :Let the $x$-axis increase from '''West''' to '''East'''. :Let the $y$-axis increase from '''South''' to '''North'''. Then the $z$-axis increases from '''below''' to '''above'''. If the $x$-axis and $y$-axis are aligned with a piece of paper or a screen aligned perpendicular to the line of sight, this translates into the following orientation: :Let the $x$-axis increase from '''left''' to '''right'''. :Let the $y$-axis increase from '''bottom''' to '''top'''. Then the $z$-axis increases from '''behind''' to '''in front''' (that is, from '''further away''' to '''closer in''').
Definition:Axis/Z-Axis
In a cartesian coordinate system, the '''$z$-axis''' is the axis passing through $x = 0, y = 0$ which is perpendicular to both the $x$-axis and the $y$-axis. It consists of all the points in the real vector space in question (usually $\R^3$) at which all the elements of its coordinates but $z$ are zero.
Definition:Formal Language
A '''formal language''' is a structure $\mathcal L$ which comprises: : A set of symbols $\mathcal A$ called the alphabet of $\mathcal L$ : A collation system with the unique readability property for $\mathcal A$ : A formal grammar that determines which collations belong to the formal language and which do not.
Definition:Coordinate System/Coordinate/Element of Ordered Pair
Let $\tuple {a, b}$ be an ordered pair. The following terminology is used: :$a$ is called the '''first coordinate''' :$b$ is called the '''second coordinate'''. This definition is compatible with the equivalent definition in the context of Cartesian coordinate systems.
Definition:Polar Coordinates
'''Polar coordinates''' are a technique for unique identification of points on the plane. A distinct point $O$ is identified. === Pole === {{:Definition:Polar Coordinates/Pole}} === Polar Axis === {{:Definition:Polar Coordinates/Polar Axis}}
Definition:Polar Coordinates/Radial Coordinate
The length of $OP$ is called the '''radial coordinate''' of $P$, and usually labelled $r$.
Definition:Polar Coordinates/Pole
The point $O$ is referred to as the '''pole''' of the polar coordinate plane.
Definition:Polar Coordinates/Polar Axis
A ray is drawn from $O$, usually to the right, and referred to as the '''polar axis'''.
Definition:Polar Coordinates/Angular Coordinate
The angle measured anticlockwise from the polar axis to $OP$ is called the '''angular coordinate''' of $P$, and usually labelled $\theta$. If the angle is measured clockwise from the polar axis to $OP$, its value is considered negative.
Definition:Meaningful Product
Let $\left({S, \circ}\right)$ be a semigroup. Let $a_1, \ldots, a_n$ be a sequence of elements of $S$. Then we define a '''meaningful product''' of $a_1, \ldots, a_n$ inductively as follows: If $n = 1$ then the only meaningful product is $a_1$. If $n > 1$ then a meaningful product is defined to be any product of the form: :$\left({a_1 \ldots a_m}\right)\left({a_{m+1} \ldots a_n}\right)$ where $m < n$ and $\left({a_1 \ldots a_m}\right)$ and  $\left({a_{m+1} \ldots a_n}\right)$ are meaningful products of $m$ and $n - m$ elements respectively.
Definition:Standard n Product
Let $\left({S, \circ}\right)$ be a semigroup. Let $a_1, \ldots, a_n$ be a sequence of elements over $S$. Then we denote the '''standard n product''' of $a_1, \ldots, a_n$ as: :$\displaystyle \prod_{i \mathop = 1}^n a_i$ We define it inductively as follows: If $n = 1$ then: :$\displaystyle \prod_{i \mathop = 1}^1 a_i = a_1$ If $n > 1$ then: :$\displaystyle \prod_{i \mathop = 1}^n a_i = \left({\displaystyle \prod_{i \mathop = 1}^{n-1} a_i}\right) a_n$
Definition:Cartesian Plane/Quadrants/First
:400px Quadrant $\text{I}: \quad$ The area above the $x$-axis and to the right of the $y$-axis is called '''the first quadrant'''. That is, '''the first quadrant''' is where both the $x$ coordinate and the $y$ coordinate of a point are positive.
Definition:Cartesian Plane/Quadrants/Third
:400px Quadrant $\text{III}: \quad$ The area below the $x$-axis and to the left of the $y$-axis is called '''the third quadrant'''. That is, '''the third quadrant''' is where both the $x$ coordinate and the $y$ coordinate of a point are negative.
Definition:Cartesian Plane/Quadrants/Fourth
:400px Quadrant $\text{IV}: \quad$ The area below the $x$-axis and to the right of the $y$-axis is called '''the fourth quadrant'''. That is, '''the fourth quadrant''' is where the $x$ coordinate of a point is positive and the $y$ coordinate of a point is negative.
Definition:Cartesian Plane/Quadrants/Second
:400px Quadrant $\text{II}: \quad$ The area above the $x$-axis and to the left of the $y$-axis is called '''the second quadrant'''. That is, '''the second quadrant''' is where the $x$ coordinate of a point is negative and the $y$ coordinate of a point is positive.
Definition:Canonical Epimorphism
Let $m \in \Z$. Let $f:\Z \to \Z_m$ be a mapping such that: :$\forall n \in \Z: \map f n = \eqclass n m$ where: ::$\Z_m$ denotes the integers modulo $m$. ::$\eqclass n m$ denotes the residue class of $n$ modulo $m$. Then $f$ is referred to as the '''canonical epimorphism''' ( '''from $\Z$ to $\Z_m$'''). That this is an epimorphism is proved in Canonical Epimorphism is Epimorphism. {{expand|Note that this definition is unnecessarily specific - it is an instance of the concept as applied to the integers modulo $m$. It may be appropriate to link to the fuller definition as given in Definition:Quotient Epimorphism, or depending on the context in Hungerford it may be better to rewrite this as a proof that it is such a specific instance. OTOH that ought already to have been documented somewhere (search around) as this area of group theory has already been covered in considerable detail.<br>In fact I've found it: Ring Epimorphism from Integers to Integers Modulo m.}}
Definition:Tangent Function
=== Definition from Triangle === {{:Definition:Tangent Function/Definition from Triangle}} === Definition from Circle === {{:Definition:Tangent Function/Definition from Circle}} === Real Function === {{:Definition:Tangent Function/Real}} === Complex Function === {{:Definition:Tangent Function/Complex}}
Definition:Tangent Function/Real
Let $x \in \R$ be a real number. The real function $\tan x$ is defined as: :$\tan x = \dfrac {\sin x} {\cos x}$ where: : $\sin x$ is the sine of $x$ : $\cos x$ is the cosine of $x$. The definition is valid for all $x \in \R$ such that $\cos x \ne 0$.
Definition:Tangent Function/Complex
Let $z \in \C$ be a complex number. The complex function $\tan z$ is defined as: :$\tan z = \dfrac {\sin z} {\cos z}$ where: : $\sin z$ is the sine of $z$ : $\cos z$ is the cosine of $z$. The definition is valid for all $z \in \C$ such that $\cos z \ne 0$.
Definition:Cotangent/Real Function
Let $x \in \R$ be a real number. The real function $\cot x$ is defined as: :$\cot x = \dfrac {\cos x} {\sin x} = \dfrac 1 {\tan x}$ where: : $\sin x$ is the sine of $x$ : $\cos x$ is the cosine of $x$ : $\tan x$ is the tangent of $x$ The definition is valid for all $x \in \R$ such that $\sin x \ne 0$.
Definition:Cotangent/Complex Function
Let $z \in \C$ be a complex number. The complex function $\cot z$ is defined as: :$\cot z = \dfrac {\cos z} {\sin z} = \dfrac 1 {\tan z}$ where: : $\sin z$ is the sine of $z$ : $\cos z$ is the cosine of $z$ : $\tan z$ is the tangent of $z$ The definition is valid for all $z \in \C$ such that $\cos z \ne 0$.
Definition:Secant Function/Real
Let $x \in \R$ be a real number. The real function $\sec x$ is defined as: :$\sec x = \dfrac 1 {\cos x}$ where $\cos x$ is the cosine of $x$. The definition is valid for all $x \in \R$ such that $\cos x \ne 0$.
Definition:Secant Function/Complex
Let $z \in \C$ be a complex number. The complex function $\sec z$ is defined as: :$\sec z = \dfrac 1 {\cos z}$ where $\cos z$ is the cosine of $z$. The definition is valid for all $z \in \C$ such that $\cos z \ne 0$.
Definition:Cosecant/Real Function
Let $x \in \C$ be a real number. The real function $\csc x$ is defined as: :$\csc x = \dfrac 1 {\sin x}$ where $\sin x$ is the sine of $x$. The definition is valid for all $x \in \R$ such that $\sin x \ne 0$.
Definition:Cosecant/Complex Function
Let $z \in \C$ be a complex number. The complex function $\csc z$ is defined as: :$\csc z = \dfrac 1 {\sin z}$ where $\sin z$ is the sine of $z$. The definition is valid for all $z \in \C$ such that $\sin z \ne 0$.
Definition:Sine/Definition from Triangle
:400px In the above right triangle, we are concerned about the angle $\theta$. The '''sine''' of $\angle \theta$ is defined as being $\dfrac {\text{Opposite}} {\text{Hypotenuse}}$.
Definition:Sine/Definition from Circle/First Quadrant
Consider a unit circle $C$ whose center is at the origin of a cartesian plane. :500px Let $P = \tuple {x, y}$ be the point on $C$ in the first quadrant such that $\theta$ is the angle made by $OP$ with the $x$-axis. Let $AP$ be the perpendicular from $P$ to the $x$-axis. Then the '''sine''' of $\theta$ is defined as the length of $AP$.
Definition:Sine/Definition from Circle/Second Quadrant
:500px Let $P = \tuple {x, y}$ be the point on $C$ in the second quadrant such that $\theta$ is the angle made by $OP$ with the $x$-axis. Let $AP$ be the perpendicular from $P$ to the $x$-axis. Then the '''sine''' of $\theta$ is defined as the length of $AP$.
Definition:Sine/Definition from Circle/Third Quadrant
:500px Let $P = \tuple {x, y}$ be the point on $C$ in the third quadrant such that $\theta$ is the angle made by $OP$ with the $x$-axis. Let $AP$ be the perpendicular from $P$ to the $x$-axis. Then the '''sine''' of $\theta$ is defined as the length of $AP$.
Definition:Sine/Definition from Circle/Fourth Quadrant
:500px Let $P = \tuple {x, y}$ be the point on $C$ in the fourth quadrant such that $\theta$ is the angle made by $OP$ with the $x$-axis. Let $AP$ be the perpendicular from $P$ to the $x$-axis. Then the '''sine''' of $\theta$ is defined as the length of $AP$.
Definition:Sine/Definition from Circle
{{:Definition:Sine/Definition from Circle/First Quadrant}}
Definition:Right Angle/Perpendicular/Foot
The '''foot''' of a perpendicular is the point where it intersects the line to which it is at right angles. In the above diagram, the point $C$ is the '''foot''' of the perpendicular $CD$.
Definition:Right Angle/Perpendicular
{{EuclidDefinition|book = I|def = 10|name = Right Angle}} :400px In the above diagram, the line $CD$ has been constructed so as to be a '''perpendicular''' to the line $AB$.
Definition:Cosine/Definition from Triangle
:400px In the above right triangle, we are concerned about the angle $\theta$. The '''cosine''' of $\angle \theta$ is defined as being $\dfrac {\text{Adjacent}} {\text{Hypotenuse}}$.
Definition:Cosine/Definition from Circle
{{:Definition:Cosine/Definition from Circle/First Quadrant}}