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<math display=block>d(p,q)\cdot d(r,s)+d(q,r)\cdot d(p,s)\ge d(p,r)\cdot d(q,s).</math>
Untuk titik-titik di bidang tersebut, rumus di atas dapat dikatakan bahwa untuk setiap [[segi empat]], perkalian antara sisi yang berhadapan dari jumlah segi empat lebih besar dari perkalian dari sisi diagonalnya. Akan tetapi, pertidaksamaan Ptolemaus lebih umumnya berlaku untuk titik-titik yang ada di ruang Euklides untuk setiap dimensi, tidak peduli bentuk susunannya.<ref>{{citation|title=Rays, Waves, and Scattering: Topics in Classical Mathematical Physics|series=Princeton Series in Applied Mathematics|first=John A.|last=Adam|publisher=Princeton University Press|year=2017|isbn=978-1-4008-8540-4|pages=26–27|chapter-url=https://books.google.com/books?id=DnygDgAAQBAJ&pg=PA26|chapter=Chapter 2. Introduction to the "Physics" of Rays|doi=10.1515/9781400885404-004}}</ref> Untuk titik-titik di ruang metrik yang bukan ruang Euklides, pertidaksamaan ini tidak berlaku benar. [[Geometri jarak]] Euklides mempelajari sifat-sifat dari jarak Euklides seperti pertidaksamaan Ptolemaus, and mempunyai penerapan yang menentukan himpunan yang diberikan dari jarak yang dimulai dari titik di ruang Euklides.<ref>{{citation|title=Euclidean Distance Geometry: An Introduction|series=Springer Undergraduate Texts in Mathematics and Technology|first1=Leo|last1=Liberti|first2=Carlile|last2=Lavor|publisher=Springer|year=2017|isbn=978-3-319-60792-4|page=xi|url=https://books.google.com/books?id=jOQ2DwAAQBAJ&pg=PP10}}</ref>
== Jarak Euklides kuadrat ==
{{multiple image
|image1=3d-function-5.svg
|caption1=A [[cone]], the [[Graph of a function|graph]] of Euclidean distance from the origin in the plane
|image2=3d-function-2.svg
|caption2=A [[paraboloid]], the graph of squared Euclidean distance from the origin
}}
Dalam banyak penerapan, khususnya saat membandingkan jarak, perhitungan akar kuadrat dalam jarak Euklides mungkin terasa lebih nyaman jika dihilangkan. Nilai dari hasil tersebut merupakan [[Kuadrat (aljabar)|penguadratan]] dari jarak Euklides, dan disebut '''jarak Euklides kuadrat'''.<ref name=spencer /> Jarak Euklides kuadrat dapat dinyatakan sebagai [[jumlah kuadrat]] melalui persamaan berikut:
<math display=block>d^2(p,q) = (p_1 - q_1)^2 + (p_2 - q_2)^2+\cdots+(p_i - q_i)^2+\cdots+(p_n - q_n)^2.</math>
Selain penerapannya dalam membandingkan jarak, jarak Euklides kuadrat merupakan alat penting di bidang [[statistika]], yang dipakai dalam metode [[kuadrat terkecil]], a standard method of fitting statistical estimates to data by minimizing the average of the squared distances between observed and estimated values,<ref>{{citation|title=Basic Statistics in Multivariate Analysis|series=Pocket Guide to Social Work Research Methods|first1=Karen A.|last1=Randolph|author1-link=Karen Randolph|first2=Laura L.|last2=Myers|publisher=Oxford University Press|year=2013|isbn=978-0-19-976404-4|page=116|url=https://books.google.com/books?id=WgSnudjEsrMC&pg=PA116}}</ref> and as the simplest form of [[divergence]] to compare [[probability distribution]]s.<ref>{{citation
| last = Csiszár | first = I. | author-link = Imre Csiszár
| doi = 10.1214/aop/1176996454
| journal = [[Annals of Probability]]
| jstor = 2959270
| mr = 365798
| pages = 146–158
| title = {{mvar|I}}-divergence geometry of probability distributions and minimization problems
| volume = 3
| year = 1975}}</ref> The addition of squared distances to each other, as is done in least squares fitting, corresponds to an operation on (unsquared) distances called [[Pythagorean addition]].<ref>{{citation |author=Moler, Cleve and Donald Morrison |title=Replacing Square Roots by Pythagorean Sums |journal=IBM Journal of Research and Development |volume=27 |issue=6 |pages=577–581 |year=1983 |url=http://www.research.ibm.com/journal/rd/276/ibmrd2706P.pdf |doi=10.1147/rd.276.0577 | citeseerx = 10.1.1.90.5651 }}</ref> In [[cluster analysis]], squared distances can be used to strengthen the effect of longer distances.<ref name=spencer>{{citation|title=Essentials of Multivariate Data Analysis|first=Neil H.|last=Spencer|publisher=CRC Press|year=2013|isbn=978-1-4665-8479-2|contribution=5.4.5 Squared Euclidean Distances|page=95|contribution-url=https://books.google.com/books?id=EG3SBQAAQBAJ&pg=PA95}}</ref>
Squared Euclidean distance does not form a metric space, as it does not satisfy the triangle inequality.<ref>{{citation|last1=Mielke|first1=Paul W.|last2=Berry|first2=Kenneth J.|editor1-last=Brown|editor1-first=Timothy J.|editor2-last=Mielke|editor2-first=Paul W. Jr.|contribution=Euclidean distance based permutation methods in atmospheric science|doi=10.1007/978-1-4757-6581-6_2|pages=7–27|publisher=Springer|title=Statistical Mining and Data Visualization in Atmospheric Sciences|year=2000}}</ref> However it is a smooth, strictly [[convex function]] of the two points, unlike the distance, which is non-smooth (near pairs of equal points) and convex but not strictly convex. The squared distance is thus preferred in [[optimization theory]], since it allows [[convex analysis]] to be used. Since squaring is a [[monotonic function]] of non-negative values, minimizing squared distance is equivalent to minimizing the Euclidean distance, so the optimization problem is equivalent in terms of either, but easier to solve using squared distance.<ref>{{citation|title=Maxima and Minima with Applications: Practical Optimization and Duality|volume=51|series=Wiley Series in Discrete Mathematics and Optimization|first=Wilfred|last=Kaplan|publisher=John Wiley & Sons|year=2011|isbn=978-1-118-03104-9|page=61|url=https://books.google.com/books?id=bAo6KNZcUP0C&pg=PA61}}</ref>
The collection of all squared distances between pairs of points from a finite set may be stored in a [[Euclidean distance matrix]], and is used in this form in distance geometry.<ref>{{citation|title=Euclidean Distance Matrices and Their Applications in Rigidity Theory|first=Abdo Y.|last=Alfakih|publisher=Springer|year=2018|isbn=978-3-319-97846-8|page=51|url=https://books.google.com/books?id=woJyDwAAQBAJ&pg=PA51}}</ref>
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