nyuuzyou/wikis huggingface dataset:
Prosthaphaeresis Formulas/Cosine minus Cosine

== Theorem ==
:$\cos \alpha - \cos \beta = -2 \, \map \sin {\dfrac {\alpha + \beta} 2} \, \map \sin {\dfrac {\alpha - \beta} 2}$
where $\sin$ denotes sine and $\cos$ denotes cosine.
== Proof 1 ==
== Proof 2 ==
== Also known as ==
== Examples ==
== Also see ==
* Sine plus Sine
* Sine minus Sine
* Cosine plus Cosine
== Linguistic Note ==
== Sources ==
* : $\text V$. Trigonometry: Formulae $(31)$
* : $\S 5$: Trigonometric Functions: $5.64$
* : $4.$ Mathematics: $4.3$ Trigonometric identities and hyperbolic functions
* 
* 
* 
* : Appendix $12$: Trigonometric formulae: Sums and differences
* : Appendix $14$: Trigonometric formulae: Factor formulae
Category:Cosine minus Cosine
Category:Cosine Function
Category:Prosthaphaeresis
Prosthaphaeresis Formulas/Cosine plus Cosine

== Theorem ==
:$\cos \alpha + \cos \beta = 2 \map \cos {\dfrac {\alpha + \beta} 2} \map \cos {\dfrac {\alpha - \beta} 2}$
where $\cos$ denotes cosine.
== Proof 1 ==
== Proof 2 ==
== Examples ==
== Also known as ==
== Also see ==
* Sine plus Sine
* Sine minus Sine
* Cosine minus Cosine
== Linguistic Note ==
== Sources ==
* : $\text V$. Trigonometry: Formulae $(30)$
* : $\S 5$: Trigonometric Functions: $5.63$
* : $4.$ Mathematics: $4.3$ Trigonometric identities and hyperbolic functions
* 
* 
* 
* : Appendix $12$: Trigonometric formulae: Sums and differences
* : Appendix $14$: Trigonometric formulae: Factor formulae
Category:Cosine plus Cosine
Category:Cosine Function
Category:Prosthaphaeresis
Prosthaphaeresis Formulas/Hyperbolic Cosine minus Hyperbolic Cosine

== Theorem ==
:$\cosh x - \cosh y = 2 \map \sinh {\dfrac {x + y} 2} \map \sinh {\dfrac {x - y} 2}$
where $\sinh$ denotes hyperbolic sine and $\cosh$ denotes hyperbolic cosine.
== Proof 1 ==
== Proof 2 ==
== Linguistic Note ==
== Sources ==
* : $\S 8$: Hyperbolic Functions: $8.45$: Sum, Difference and Product of Hyperbolic Functions
Category:Hyperbolic Cosine minus Hyperbolic Cosine
Category:Hyperbolic Cosine Function
Category:Prosthaphaeresis
Prosthaphaeresis Formulas/Hyperbolic Cosine plus Hyperbolic Cosine

== Theorem ==
:$\cosh x + \cosh y = 2 \map \cosh {\dfrac {x + y} 2} \map \cosh {\dfrac {x - y} 2}$
where $\cosh$ denotes hyperbolic cosine.
== Proof ==
== Linguistic Note ==
== Sources ==
* : $\S 8$: Hyperbolic Functions: $8.44$: Sum, Difference and Product of Hyperbolic Functions
Category:Hyperbolic Cosine Function
Category:Prosthaphaeresis
Prosthaphaeresis Formulas/Hyperbolic Sine minus Hyperbolic Sine

== Theorem ==
:$\sinh x - \sinh y = 2 \map \cosh {\dfrac {x + y} 2} \map \sinh {\dfrac {x - y} 2}$
where $\sinh$ denotes hyperbolic sine and $\cosh$ denotes hyperbolic cosine.
== Proof ==
== Linguistic Note ==
== Sources ==
* : $\S 8$: Hyperbolic Functions: $8.43$: Sum, Difference and Product of Hyperbolic Functions
Category:Hyperbolic Sine Function
Category:Prosthaphaeresis
Prosthaphaeresis Formulas/Hyperbolic Sine plus Hyperbolic Sine

== Theorem ==
:$\sinh x + \sinh y = 2 \map \sinh {\dfrac {x + y} 2} \map \cosh {\dfrac {x - y} 2}$
where $\sinh$ denotes hyperbolic sine and $\cosh$ denotes hyperbolic cosine.
== Proof ==
== Linguistic Note ==
== Sources ==
* : $\S 8$: Hyperbolic Functions: $8.42$: Sum, Difference and Product of Hyperbolic Functions
Category:Hyperbolic Sine Function
Category:Prosthaphaeresis
Prosthaphaeresis Formulas/Sine minus Sine

== Theorem ==
:$\sin \alpha - \sin \beta = 2 \map \cos {\dfrac {\alpha + \beta} 2} \map \sin {\dfrac {\alpha - \beta} 2}$
where $\sin$ denotes sine and $\cos$ denotes cosine.
== Proof 1 ==
== Proof 2 ==
== Also known as ==
== Examples ==
== Also see ==
* Sine plus Sine
* Cosine plus Cosine
* Cosine minus Cosine
== Linguistic Note ==
== Sources ==
* : $\text V$. Trigonometry: Formulae $(29)$
* : $\S 5$: Trigonometric Functions: $5.62$
* : $4.$ Mathematics: $4.3$ Trigonometric identities and hyperbolic functions
* 
* 
* 
* : Appendix $12$: Trigonometric formulae: Sums and differences
* : Appendix $14$: Trigonometric formulae: Factor formulae
Category:Sine minus Sine
Category:Sine Function
Category:Prosthaphaeresis
Prosthaphaeresis Formulas/Sine plus Sine

== Theorem ==
:$\sin \alpha + \sin \beta = 2 \map \sin {\dfrac {\alpha + \beta} 2} \map \cos {\dfrac {\alpha - \beta} 2}$
where $\sin$ denotes sine and $\cos$ denotes cosine.
== Proof 1 ==
== Proof 2 ==
== Also reported as ==
This result is also sometimes reported as:
:$\dfrac {\sin \alpha + \sin \beta} 2 = \map \sin {\dfrac {\alpha + \beta} 2} \map \cos {\dfrac {\alpha - \beta} 2}$
== Also known as ==
== Examples ==
== Also see ==
* Sine minus Sine 
* Cosine plus Cosine
* Cosine minus Cosine
== Linguistic Note ==
== Sources ==
* : $\text V$. Trigonometry: Formulae $(28)$
* : $\S 5$: Trigonometric Functions: $5.61$
* : $4.$ Mathematics: $4.3$ Trigonometric identities and hyperbolic functions
* 
* 
* : Chapter $5$: Eternal Triangles: Napierian logarithms
* 
* : Appendix $12$: Trigonometric formulae: Sums and differences
* : Appendix $14$: Trigonometric formulae: Factor formulae
Category:Sine plus Sine
Category:Prosthaphaeresis
Proth Prime/Examples/8423 x 2^59,877 + 1

== Example of Proth Prime ==
The Proth number:
:$8423 \times 2^{59 \, 877} + 1$
is a Proth prime.
== Proof ==
== Historical Note ==
== Sources ==
* 
* 
* : $8423 \times 2^{59,877} + 1$
Category:Proth Primes
Proth Prime/Examples/8423 x 2^59,877 + 1/Historical Note

== Historical Note on Proth Prime $8423 \times 2^{59 \, 877} + 1$ ==
The Proth number $8423 \times 2^{59 \, 877} + 1$ was identified as being a Proth prime in $1988$ by and .
== Sources ==
* 
* : $8423 \times 2^{59,877} + 1$
Category:Historical Notes
Proton has Smallest Positive Electric Charge Possible

== Physical Law ==
The electric charge on the proton is the negative charge with the smallest possible magnitude that can be directly observed.
== Justification ==
Electric charge has been observed as being quantized.
The quantum of positive charge is defined as being the elementary charge.
The proton bears exactly that elementary charge.
== Sources ==
* : Chapter $\text {I}$: Preliminary Survey: $\S 1$. Electrostatics
Category:Electric Charge
Category:Protons
Proton to Electron Mass Ratio

== Physical Law ==
The ratio of the mass of the proton to the mass of an electron is given by:
:$\dfrac {m_{\mathrm p} } {m_{\mathrm e} } \approx 1 836 \cdotp 15267 \,343$
== Demonstration ==
From Mass of Proton, we have:
:$m_{\mathrm p} \approx 1.67262 \, 19236 \, 9(51) \times 10^{-27}$ kilograms
From Mass of Electron, we have:
:$m_{\mathrm e} \approx 9.10938 \, 370 \, 15(28) \times 10^{-31}$ kilograms
Hence:
:$\dfrac {m_{\mathrm p} } {m_{\mathrm e} } \approx 1 836 \cdotp 15267 \,343$
== Sources ==
* : Chapter $\text {I}$: Preliminary Survey: $\S 1$. Electrostatics
* : $3.$ Physical Constants
Category:Electrons
Category:Protons
Category:Mass
Provable Consequence of Theorems is Theorem

== Theorem ==
Let $\PP$ be a proof system for a formal language $\LL$.
Let $\FF$ be a collection of theorems of $\PP$.
Denote with $\map {\mathscr P} \FF$ the proof system obtained from $\mathscr P$ by adding all the WFFs from $\FF$ as axioms.
Let $\phi$ be a provable consequence of $\FF$:
:$\vdash_{\mathscr P} \FF$
:$\FF \vdash_{\mathscr P} \phi$
Then $\phi$ is also a theorem of $\mathscr P$:
:$\vdash_{\mathscr P} \phi$
== Proof ==
We have that $\phi$ is a provable consequence of $\FF$.
Hence it is a theorem of $\map {\mathscr P} \FF$, the proof system obtained from $\mathscr P$ by adding all of $\FF$ as axioms.
Now in the formal proof of $\phi$ in $\map {\mathscr P} \FF$, both axioms and rules of inference are used.
Each rule of inference of $\map {\mathscr P} \FF$ is also a rule of inference of $\mathscr P$.
Similarly, by construction, each axiom of $\map {\mathscr P} \FF$ is either an axiom of $\mathscr P$ or an element of $\FF$.
But the elements of $\FF$ are theorems of $\mathscr P$, each of which thus has a formal proof in $\mathscr P$.
By definition, the rules of inference of a proof system do not distinguish between theorems and axioms.
Therefore the formal proofs of the following can be combined:
:$\vdash_{\mathscr P} \FF$
:$\vdash_{\map {\mathscr P} \FF} \phi$
by letting the latter follow the former.
This yields a new formal proof, which is entirely formulated in $\mathscr P$.
This is the desired formal proof of $\phi$ from $\mathscr P$, and we conclude:
:$\vdash_{\mathscr P} \phi$
== Sources ==
* : $\text{II}.11$ Some Strategies for Constructing Proofs: Lemma $\text{II}.11.7$
Category:Proof Systems
Provable by Gentzen Proof System iff Negation has Closed Tableau/Formula

== Theorem ==
Let $\mathscr G$ be instance $1$ of a Gentzen proof system.
Let $\mathbf A$ be a WFF of propositional logic.
Then $\mathbf A$ is a $\mathscr G$-theorem :
:$\neg \mathbf A$ has a closed semantic tableau
where $\neg \mathbf A$ is the negation of $\mathbf A$.
== Proof ==
This is a specific instance of Provable by Gentzen Proof System iff Negation has Closed Tableau: Set of Formulas.
== Sources ==
* : $\S 3.2.1$: Theorem $3.6$
Category:Propositional Logic
Pseudocompact Normal Space is Countably Compact

== Theorem ==
Let $T = \struct {S, \tau}$ be a normal space.
Then $T$ is pseudocompact $T$ is countably compact.
== Proof ==
Let $T = \struct {S, \tau}$ be a normal space.
By Countably Compact Space is Pseudocompact we already have that if $T$ is countably compact then $T$ is pseudocompact.
It remains to prove that if $T$ is pseudocompact then $T$ is countably compact.
 $T$ is not countably compact.
By definition of countably compact space, $S$ contains an infinite subset $H = \set {x_n}$ with no $\omega$-accumulation point in $T$.
Since, by definition of normal, $S$ is a $T_1$ space, $H$ is closed and discrete in the subspace topology.
By definition, $T$ is pseudocompact every continuous real-valued function on $S$ is bounded.
By definition of normal, we have that $S$ is a $T_4$ space.
Hence the Tietze Extension Theorem guarantees a continuous extension to $S$ of the unbounded continuous mapping $f: H \to \R$ defined by $\map f {x_n} = n$.
This contradicts our hypothesis that $S$ is pseudocompact.
It follows by Proof by Contradiction that $T$ is countably compact.
== Sources ==
* : Part $\text I$: Basic Definitions: Section $3$: Compactness: Global Compactness Properties
Category:Countably Compact Spaces
Category:Pseudocompact Spaces
Category:Normal Spaces
Pseudocomplemented Lattice is Bounded

== Theorem ==
Let $\struct {L, \wedge, \vee, \preceq}$ be a pseudocomplemented lattice.
Then $\struct {L, \wedge, \vee, \preceq}$ is a bounded lattice.
== Proof ==
By the definition of pseudocomplemented lattice, $L$ has a smallest element $\bot$.
Let $x \in L$.
Then:
:$x \wedge \bot = \bot$
By the definition of pseudocomplemented lattice, there is a greatest element $\bot^*$ such that:
:$\bot \wedge \bot^* = \bot$
But then by the definition of greatest element:
:$\forall x \in L: x \preceq \bot^*$
Hnce.o $\bot^*$ is the greatest element of $L$.
Since $L$ has a smallest element and a greatest element, it is a bounded lattice.
Category:Pseudocomplemented Lattices
Category:Bounded Lattices
Pseudometric Defines an Equivalence Relation

== Theorem ==
Let $X$ be a set on which there is a pseudometric $d: X \times X \to \R$.
For any $x, y \in X$, let $x \sim y$ denote that $\map d {x, y} = 0$.
Then $\sim$ is an equivalence relation, and the equivalence classes consist of sets of elements of $X$ at zero distance from each other.
== Proof ==
Checking in turn each of the criteria for equivalence:
=== Reflexive ===
From the definition, we have that $\forall x \in X: \map d {x, x} = 0$.
So $\sim$ is a reflexive relation.
=== Symmetric ===
From the definition, we have that $\forall x, y \in X: \map d {x, y} = \map d {y, x}$.
So $\sim$ is a symmetric relation.
=== Transitive ===
From the definition, we have that $\map d {x, y} + \map d {y, z} \ge \map d {x, z}$.
So if $\map d {x, y} = 0$ and $\map d {y, z} = 0$ it follows directly that $\map d {x, z} = 0$.
So $\sim$ is a transitive relation.
$\sim$ has been shown to be reflexive, symmetric and transitive, and so by definition an equivalence relation.
Category:Examples of Equivalence Relations
Category:Pseudometric Spaces
Pseudometric Induced by Seminorm is Pseudometric

== Definition ==
Let $\GF \in \set {\R, \C}$. 
Let $X$ be a vector space over $\GF$. 
Let $p$ be a seminorm on $X$.
Let $d_p$ be the pseudometric induced by $p$. 
Then $d_p$ is a pseudometric.
== Proof ==
=== Proof of ===
For each $x \in X$ we have: 
:$\map {d_p} {x, x} = \map p {x - x} = \map p { {\mathbf 0}_X}$
From Seminorm Maps Zero Vector to Zero, we therefore have: 
:$\map {d_p} {x, x} = 0$
=== Proof of ===
For $x, y, z \in X$ we have: 
=== Proof of ===
For $x, y \in X$ we have: 
Category:Pseudometric Spaces
Category:Seminorms
Pseudometric Space generates Uniformity

== Theorem ==
Let $P = \struct {A, d}$ be a pseudometric space.
Let $\UU$ be the set of sets defined as:
:$\UU := \set {u \in A \times A: \exists \epsilon \in \R_{>0}: u_\epsilon \subseteq u}$
where:
:$\R_{>0}$ is the set of strictly positive real numbers
:$u_\epsilon$ is defined as:
::$u_\epsilon := \set {\tuple {x, y}: \map d {x, y} < \epsilon}$
Then $\UU$ is a uniformity on $A$ which generates a uniform space with the same topology as the topology induced by $d$.
== Proof ==
We check whether the Uniformity Axioms are satisfied.
=== $\text U 1$ ===
Let $x \in A$.
Then $\tuple {x, x} \in \Delta_A$.
Let $u \in \UU$.
From definition:
:$\exists \epsilon \in \R_{>0}: u_\epsilon \subseteq u$
By :
:$\map d {x, x} = 0 < \epsilon$
Hence we have $\tuple {x, x} \in u_\epsilon$.
By definition of subset, $\Delta_A \subseteq u_\epsilon \subseteq u$.
=== $\text U 2$ ===
Let $u, v \in \UU$.
From definition:
:$\exists \epsilon \in \R_{>0}: u_\epsilon \subseteq u$
:$\exists \delta \in \R_{>0}: u_\delta \subseteq v$
 assume that $\delta \ge \epsilon$.
Let $\tuple {x, y} \in u_\epsilon$.
Then $\map d {x, y} < \epsilon \le \delta$.
Thus $\tuple {x, y} \in u_\delta$.
By definition of subset, $u_\epsilon \subseteq u_\delta$.
Hence:
=== $\text U 3$ ===
Let $v \in A \times A$.
Suppose $u \in \UU$ and $u \subseteq v$.
From definition:
:$\exists \epsilon \in \R_{>0}: u_\epsilon \subseteq u$
Thus $u_\epsilon \subseteq v$.
Therefore $v \in \UU$ as well.
=== $\text U 4$ ===
Let $u \in \UU$.
From definition:
:$\exists \epsilon \in \R_{>0}: u_\epsilon \subseteq u$
Let $v = u_{\epsilon / 2}$.
Since $u_{\epsilon / 2} \subseteq v$ and $\dfrac \epsilon 2 > 0$, $v \in \UU$.
From definition:
Let $\tuple {x, z} \in v \circ v$.
Then:
:$\exists y \in A: \map d {x, y} < \dfrac \epsilon 2, \map d {y, z} < \dfrac \epsilon 2$
By :
Hence $\tuple {x, z} \in u_\epsilon \subseteq u$.
By definition of subset, $v \circ v \subseteq u$.
=== $\text U 5$ ===
Let $u \in \UU$.
From definition:
:$\exists \epsilon \in \R_{>0}: u_\epsilon \subseteq u$
Suppose $\tuple {b, a} \in u_\epsilon$.
Then $\map d {b, a} < \epsilon$.
By :
:$\map d {a, b} = \map d {b, a} < \epsilon$
Hence $\tuple {a, b} \in u_\epsilon \subseteq u$.
Consider the set $u^{-1} = \set {\tuple {y, x}: \tuple {x, y} \in u}$.
From the above, $\tuple {b, a} \in u^{-1}$.
By definition of subset, $u_\epsilon \subseteq u^{-1}$.
Hence $u^{-1} \in \UU$.
== Sources ==
* : Part $\text I$: Basic Definitions: Section $5$: Metric Spaces: Metric Uniformities
Category:Pseudometric Spaces
Category:Uniformities
Pseudometric Space is First-Countable

== Theorem ==
Let $M = \struct {A, d}$ be a pseudometric space.
Then $M$ is first-countable.
== Proof ==
Let $x \in A$.
Let:
:$\BB = \set {\map {B_{1/n} } x: n \in \N_{>0} }$
where $\map {B_\epsilon} x$ denotes the open $\epsilon$-ball of $x$ in $M$.
By Surjection from Natural Numbers iff Countable, we have that $\BB$ is countable.
By the definition of a first-countable space, it suffices to show that $\BB$ is a local basis at $x$.
By Open Ball of Pseudometric Space is Open Set, every element of $\BB$ is an open neighborhood of $x$.
Let $U$ be an open neighborhood of $x$.
By the definition of an open set, there exists a strictly positive real number $\epsilon$ such that $\map {B_\epsilon} x \subseteq U$.
By the Axiom of Archimedes, there exists a natural number $n > \dfrac 1 \epsilon$.
That is, $\dfrac 1 n < \epsilon$, and so:
:$\map {B_{1/n} } x \subseteq \map {B_\epsilon} x \subseteq U$
The result follows from Subset Relation is Transitive.
Category:Pseudometric Spaces
Category:First-Countable Spaces
Pseudometric Space is Metric Space iff Kolmogorov

== Theorem ==
Let $M = \struct {S, d}$ be a pseudometric space.
Let $T = \struct {S, \tau}$ be the topological space over $S$ induced by $d$.
Then $M$ is a metric space $T$ is a $T_0$ (Kolmogorov} space.
== Proof ==
=== Necessary Condition ===
Let $M$ be a metric space.
From Metric Space is Hausdorff, $M$ is a Hausdorff space.
From:
:$T_2$ (Hausdorff) Space is $T_1$ Space
and:
:$T_1$ Space is $T_0$ (Kolmogorov) Space
it follows that $M$ is a Kolmogorov space.
=== Sufficient Condition ===
Suppose now that $M$ is a Kolmogorov space.
Let $a, b \in S$ such that $a \ne b$.
Then $\exists U \in \tau$ such that:
:$\paren {a \in U \land b \notin U} \lor \paren {b \in U \land a \notin U}$
, suppose that $a \in U$ and $b \notin U$.
Then by the definition of the induced topology:
:$\exists \epsilon \in \R_{>0}: \map {B_\epsilon} a \subset U$
where $\map {B_\epsilon} a$ denotes the open $\epsilon$-ball of $a$ in $M$.
Since $b \notin U$:
:$\map d {a, b} \ge \epsilon$
Since $\epsilon > 0$:
:$\map d {a, b} > 0$
Since this holds for any pair of distinct points, $M$ is a metric space.
Category:Metric Spaces
Category:Pseudometric Spaces
Category:T0 Spaces
Pseudometric induces Topology

== Theorem ==
Let $S \ne \O$ be a non-empty set.
Consider a pseudometric space $\struct {S, d}$ where $d: S \times S \to \R_{\ge 0}$ is a pseudometric.
Then $\struct {S, d}$ gives rise to a topological space $\struct {S, \tau_d}$ whose topology $\tau_d$ is defined (or induced) by $d$.
== Proof ==
Let $\tau_d$ be the set of all $X \subseteq S$ which are open in the sense that:
:$\forall y \in X: \exists \epsilon > 0: \map {B_\epsilon} y \subseteq X$
where $\map {B_\epsilon} y$ is the open $\epsilon$-ball of $y$.
Equivalently:
:$\forall x \in X: \exists \epsilon \in \R_{>0}: \forall y \in S: \map d {x, y} < \epsilon \implies y \in X$
We need to show that $\tau_d$ forms a topology on $S$.
We examine each of the open set axioms in turn.
=== ===
Let $\family {U_i}_{i \mathop \in I}$ be an indexed family of open sets of $S$.
Let $\ds V = \bigcup_{i \mathop \in I} U_i$ be the union of $\family {U_i}_{i \mathop \in I}$.
Then by the definition of union:
:$\forall x \in V: \exists i \in I: x \in U_i$
and so by the definition of open set:
:$\exists \epsilon \in \R_{>0}: \map {B_\epsilon} x \subseteq U_i \subseteq V$
Hence $V$ is open by definition.
=== ===
Let $U$ and $V$ be open sets of $S$.
Let $x \in U \cap V$.
Then:
:$\exists \epsilon_U \in \R_{>0}: \map {B_{\epsilon_U} } x \subseteq U$
:$\exists \epsilon_V \in \R_{>0}: \map {B_{\epsilon_V} } x \subseteq V$
Let $\epsilon := \min \set {\epsilon_U, \epsilon_V}$.
Then:
:$\map {B_\epsilon} x \subseteq U \cap V$
Hence $U \cap V$ is open by definition.
=== ===
From Open Sets in Pseudometric Space:
:$\O \in \tau_d$
and:
:$S \in \tau_d$
All the open set axioms are fulfilled, and the result follows.
== Also see ==
* Pseudometrizable: any topological space which is homeomorphic to such a $\struct {S, \tau_d}$.
== Sources ==
* : Part $\text I$: Basic Definitions: Section $5$: Metric Spaces: Metric Uniformities
Category:Topology
Category:Pseudometric Spaces
Pseudoprime Element is Prime in Arithmetic Lattice

== Theorem ==
Let $L = \struct {S, \vee, \wedge, \preceq}$ be a bounded below arithmetic lattice.
Let $p \in S$.
Then if $p$ is pseudoprime element, then $p$ is prime element.
== Proof ==
By Arithmetic iff Way Below Relation is Multiplicative in Algebraic Lattice:
:$\ll$ is a multiplicative relation.
Thus by Way Below Relation is Multiplicative implies Pseudoprime Element is Prime:
:the result holds.
== Sources ==
* 
* 
Category:Prime Elements
Category:Continuous Lattices
Ptolemy's Theorem

== Theorem ==
Let $ABCD$ be a cyclic quadrilateral.
Then:
:$AB \times CD + AD \times BC = AC \times BD$
== Proof 1 ==
== Proof 2 ==
== Proof 3 ==
== Sources ==
* 
* 
* : Chapter $5$: Eternal Triangles: Ptolemy
* 
Category:Ptolemy's Theorem
Category:Cyclic Quadrilaterals
Ptolemy's Theorem/Proof 1

== Theorem ==
== Proof ==
:
Let $ABCD$ be a cyclic quadrilateral.
By Angles in Same Segment of Circle are Equal:
:$\angle BAC = \angle BDC$
and:
:$\angle ADB = \angle ACB$
By Construction of Equal Angle, construct $E$ on $AC$ such that:
:$\angle ABE = \angle CBD$
Since:
:$\angle ABE + \angle CBE = \angle ABC = \angle CBD + \angle ABD$
it follows that:
:$\angle CBE = \angle ABD$
By Equiangular Triangles are Similar:
:$\triangle ABE$ is similar to $\triangle DBC$
and:
:$\triangle ABD$ is similar to $\triangle EBC$
Thus:
:$\dfrac {AE} {AB} = \dfrac {CD} {BD}$
and:
:$\dfrac {CE} {BC} = \dfrac {DA} {BD}$
Equivalently:
:$AE \times BD = AB \times CD$
and:
:$CE \times BD = BC \times DA$
Adding:
:$AE \times BD + CE \times BD = AB \times CD + BC \times DA$
Factorizing:
:$\paren {AE + CE} \times BD = AB \times CD + BC \times DA$
But:
:$AE + CE = AC$
so:
:$AC \times BD = AB \times CD + BC \times DA$
Category:Ptolemy's Theorem
Ptolemy's Theorem/Proof 2

== Theorem ==
== Proof ==
Let $\Box ABCD$ be a cyclic quadrilateral, with diagonals $AC$ and $BD$. 
By Opposite Angles of Cyclic Quadrilateral sum to Two Right Angles:
:$\angle ABC$ is supplementary to $\angle ADC$
As well:
:$\angle BAD$ is supplementary to $\angle BCD$
Construct two triangles $\triangle A'B'C'$ and $\triangle C'D'E'$ congruent to $\triangle ABC$ and $\triangle CDE$ respectively, with $B'C'D'$ collinear.
$E$ is the same point that was $A$ in Figure $1$.
Hence:
:$ED = AD$
$\angle ABC$ and $\angle CDE$ are supplementary.
By Equal Corresponding Angles or Supplementary Interior Angles implies Parallel Lines:
:$AB \parallel ED$
By construction:
:$AC = EC$
Now, scale the sides of $\triangle CDE$ by the length of $AB$.
Also scale the sides of $\triangle ABC$ by the length of $DE$.
Let the scaled figure be $A'B'F'G'$.
:$A'B' = AB \cdot DE = F'G'$
By Quadrilateral is Parallelogram iff One Pair of Opposite Sides is Equal and Parallel:
:$\Box A'B'F'G'$ is a parallelogram
By construction:
:$\angle A'C'B' + \angle A'C'F'$ has the same measure of angle as the original $\angle BCD$.
Therefore the supplementary angle $\angle A'C'G'$ has the same measure as the supplementary angle $\angle BAD$. 
By construction:
:$A'C' = AC \times DE$
:$B'C' = BC \times DE$
Also by construction:
:$C'G' = CE \times AC = AD \cdot AC$
:$C'F' = CD \times AC$
Therefore $A'C'$ and $C'G'$ are in proportion with scale factor $AC$.
By Triangles with One Equal Angle and Two Sides Proportional are Similar:
:$\triangle A'C'H' \sim \triangle ABD$
Thus, the length of $A'G'$ of the new construct is equal to $BD$ times the scale factor $AC$ Euclid:Proposition/VI/4:
:$A'G' = AC \times BD$
And:
:$B'F' = B'C' + C'F'$
Substituting:
:$B'F' = BC \times DE + CD \times AC$
But by Opposite Sides and Angles of Parallelogram are Equal: 
:$A'G' = B'C'F' = B'F'$
Therefore 
:$AC \times BD = BC \times DE + CD \times AC$
Category:Ptolemy's Theorem
Ptolemy's Theorem/Proof 3

== Theorem ==
== Proof ==
Let an arbitrary circle $K$ be drawn in the plane.
Let $A$, $B$, $C$, and $D$ be arbitrary points on $K$.
By definition, $\Box ABCD$ is a cyclic quadrilateral.
We are to show that $AB \cdot CD + BC \cdot AD = AC \cdot BD$.
Let $T$ be an inversive transformation such that:
:the inversion center of $T$ is $D$
:the inversion circle $O$ for $T$ is chosen such that $K$ is completely inside $O$
:the radius of $O$ is $r$.
Let $A'$, $B'$, and $C'$ be the images of $A$, $B$, and $C$ under $T$ respectively.
By Inverse of Circle Through Inversion Center is Straight Line Not Through Inversion Center:
:$A'$, $B'$ and $C'$ are collinear.
From $(1)$ we establish via Triangles with Proportional Sides are Similar that:
and:
Similarly:
and noting the collinearity of $A'$, $B'$ and $C'$:
Finally: 
The result follows.
Category:Ptolemy's Theorem