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69-434: Perpendicularity is one particular instance of the more general mathematical concept of orthogonality ; perpendicularity is the orthogonality of classical geometric objects. Thus, in advanced mathematics, the word "perpendicular" is sometimes used to describe much more complicated geometric orthogonality conditions, such as that between a surface and its normal vector . A line is said to be perpendicular to another line if

138-413: A and b are parallel, any of the following conclusions leads to all of the others: In geometry , the perpendicular distance between two objects is the distance from one to the other, measured along a line that is perpendicular to one or both. The distance from a point to a line is the distance to the nearest point on that line. That is the point at which a segment from it to the given point

207-400: A circle is perpendicular to the tangent line to that circle at the point where the diameter intersects the circle. A line segment through a circle's center bisecting a chord is perpendicular to the chord. If the intersection of any two perpendicular chords divides one chord into lengths a and b and divides the other chord into lengths c and d , then a + b + c + d equals

276-447: A hyperbola is perpendicular to the conjugate axis and to each directrix. The product of the perpendicular distances from a point P on a hyperbola or on its conjugate hyperbola to the asymptotes is a constant independent of the location of P. A rectangular hyperbola has asymptotes that are perpendicular to each other. It has an eccentricity equal to 2 . {\displaystyle {\sqrt {2}}.} The legs of

345-404: A quadrilateral is a perpendicular to a side through the midpoint of the opposite side. An orthodiagonal quadrilateral is a quadrilateral whose diagonals are perpendicular. These include the square , the rhombus , and the kite . By Brahmagupta's theorem , in an orthodiagonal quadrilateral that is also cyclic , a line through the midpoint of one side and through the intersection point of

414-406: A right triangle are perpendicular to each other. The altitudes of a triangle are perpendicular to their respective bases . The perpendicular bisectors of the sides also play a prominent role in triangle geometry. The Euler line of an isosceles triangle is perpendicular to the triangle's base. The Droz-Farny line theorem concerns a property of two perpendicular lines intersecting at

483-472: A , g , and n ) versions of 802.11 Wi-Fi ; WiMAX ; ITU-T G.hn , DVB-T , the terrestrial digital TV broadcast system used in most of the world outside North America; and DMT (Discrete Multi Tone), the standard form of ADSL . In OFDM, the subcarrier frequencies are chosen so that the subcarriers are orthogonal to each other, meaning that crosstalk between the subchannels is eliminated and intercarrier guard bands are not required. This greatly simplifies

552-561: A line segment A B ¯ {\displaystyle {\overline {AB}}} is perpendicular to a line segment C D ¯ {\displaystyle {\overline {CD}}} if, when each is extended in both directions to form an infinite line, these two resulting lines are perpendicular in the sense above. In symbols, A B ¯ ⊥ C D ¯ {\displaystyle {\overline {AB}}\perp {\overline {CD}}} means line segment AB

621-423: A mono signal, equivalent to a stereo signal in which both channels carry identical (in-phase) signals. Thales%27s theorem In geometry , Thales's theorem states that if A , B , and C are distinct points on a circle where the line AC is a diameter , the angle ∠ ABC is a right angle . Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved as part of

690-440: A parallelogram by construction (as opposite sides are parallel). Since in a parallelogram adjacent angles are supplementary (add to 180°) and ∠ ABC is a right angle (90°) then angles ∠ BAD , ∠ BCD , ∠ ADC are also right (90°); consequently ABCD is a rectangle. Let O be the point of intersection of the diagonals AC and BD . Then the point O , by the second fact above, is equidistant from A , B , and C . And so O

759-496: A right angle ∠ ABC and circle M with AC as a diameter. Let M's center lie on the origin, for easier calculation. Then we know It follows This means that A and B are equidistant from the origin, i.e. from the center of M . Since A lies on M , so does B , and the circle M is therefore the triangle's circumcircle. The above calculations in fact establish that both directions of Thales's theorem are valid in any inner product space . As stated above, Thales's theorem

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828-469: A right angle or something related to a right angle. In mathematics , orthogonality is the generalization of the geometric notion of perpendicularity to the linear algebra of bilinear forms . Two elements u and v of a vector space with bilinear form B {\displaystyle B} are orthogonal when B ( u , v ) = 0 {\displaystyle B(\mathbf {u} ,\mathbf {v} )=0} . Depending on

897-487: A time axis determined by a rapidity of motion is hyperbolic-orthogonal to a space axis of simultaneous events, also determined by the rapidity. The theory features relativity of simultaneity . In quantum mechanics , a sufficient (but not necessary) condition that two eigenstates of a Hermitian operator , ψ m {\displaystyle \psi _{m}} and ψ n {\displaystyle \psi _{n}} , are orthogonal

966-415: A triangle in a circle where AB is a diameter in that circle. Then construct a new triangle △ ABD by mirroring △ ABC over the line AB and then mirroring it again over the line perpendicular to AB which goes through the center of the circle. Since lines AC and BD are parallel , likewise for AD and CB , the quadrilateral ACBD is a parallelogram . Since lines AB and CD , the diagonals of

1035-400: A triangle's orthocenter . Harcourt's theorem concerns the relationship of line segments through a vertex and perpendicular to any line tangent to the triangle's incircle . In a square or other rectangle , all pairs of adjacent sides are perpendicular. A right trapezoid is a trapezoid that has two pairs of adjacent sides that are perpendicular. Each of the four maltitudes of

1104-448: Is time-division multiple access (TDMA), where the orthogonal basis functions are nonoverlapping rectangular pulses ("time slots"). Another scheme is orthogonal frequency-division multiplexing (OFDM), which refers to the use, by a single transmitter, of a set of frequency multiplexed signals with the exact minimum frequency spacing needed to make them orthogonal so that they do not interfere with each other. Well known examples include (

1173-617: Is a point on the unit circle (cos θ , sin θ ) . We will show that △ ABC forms a right angle by proving that AB and BC are perpendicular — that is, the product of their slopes is equal to −1. We calculate the slopes for AB and BC : Then we show that their product equals −1: Note the use of the Pythagorean trigonometric identity sin 2 ⁡ θ + cos 2 ⁡ θ = 1. {\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1.} Let △ ABC be

1242-425: Is a special case of the inscribed angle theorem (the proof of which is quite similar to the first proof of Thales's theorem given above): A related result to Thales's theorem is the following: Thales's theorem can be used to construct the tangent to a given circle that passes through a given point. In the figure at right, given circle k with centre O and the point P outside k , bisect OP at H and draw

1311-464: Is a strategy allowing the deprotection of functional groups independently of each other. In supramolecular chemistry the notion of orthogonality refers to the possibility of two or more supramolecular, often non-covalent , interactions being compatible; reversibly forming without interference from the other. In analytical chemistry , analyses are "orthogonal" if they make a measurement or identification in completely different ways, thus increasing

1380-452: Is also perpendicular to the first. For this reason, we may speak of two lines as being perpendicular (to each other) without specifying an order. A great example of perpendicularity can be seen in any compass, note the cardinal points; North, East, South, West (NESW) The line N-S is perpendicular to the line W-E and the angles N-E, E-S, S-W and W-N are all 90° to one another. Perpendicularity easily extends to segments and rays . For example,

1449-417: Is called an orthogonal map. In philosophy , two topics, authors, or pieces of writing are said to be "orthogonal" to each other when they do not substantively cover what could be considered potentially overlapping or competing claims. Thus, texts in philosophy can either support and complement one another, they can offer competing explanations or systems, or they can be orthogonal to each other in cases where

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1518-410: Is called the perpendicular bisector of the line segment connecting the points. The perpendicular bisectors of any two sides of a triangle intersect in exactly one point. This point must be equidistant from the vertices of the triangle.) This circle is called the circumcircle of the triangle. One way of formulating Thales's theorem is: if the center of a triangle's circumcircle lies on the triangle then

1587-530: Is center of the circumscribing circle, and the hypotenuse of the triangle ( AC ) is a diameter of the circle. Given a right triangle ABC with hypotenuse AC , construct a circle Ω whose diameter is AC . Let O be the center of Ω . Let D be the intersection of Ω and the ray OB . By Thales's theorem, ∠ ADC is right. But then D must equal B . (If D lies inside △ ABC , ∠ ADC would be obtuse, and if D lies outside △ ABC , ∠ ADC would be acute.) This proof utilizes two facts: Let there be

1656-416: Is easier to verify designs that neither cause side effects nor depend on them. An instruction set is said to be orthogonal if it lacks redundancy (i.e., there is only a single instruction that can be used to accomplish a given task) and is designed such that instructions can use any register in any addressing mode . This terminology results from considering an instruction as a vector whose components are

1725-411: Is equidistant from the vertices and so is the center of the circumscribing circle of the original triangle, it utilizes two facts: Let there be a right angle ∠ ABC , r a line parallel to BC passing by A , and s a line parallel to AB passing by C . Let D be the point of intersection of lines r and s . (It has not been proven that D lies on the circle.) The quadrilateral ABCD forms

1794-414: Is exemplified in the top diagram, above, and its caption. The diagram can be in any orientation. The foot is not necessarily at the bottom. More precisely, let A be a point and m a line. If B is the point of intersection of m and the unique line through A that is perpendicular to m , then B is called the foot of this perpendicular through A . To make the perpendicular to the line AB through

1863-408: Is perpendicular to a second line, it is also perpendicular to any line parallel to that second line. In the figure at the right, all of the orange-shaded angles are congruent to each other and all of the green-shaded angles are congruent to each other, because vertical angles are congruent and alternate interior angles formed by a transversal cutting parallel lines are congruent. Therefore, if lines

1932-428: Is perpendicular to line segment CD. A line is said to be perpendicular to a plane if it is perpendicular to every line in the plane that it intersects. This definition depends on the definition of perpendicularity between lines. Two planes in space are said to be perpendicular if the dihedral angle at which they meet is a right angle. The word foot is frequently used in connection with perpendiculars. This usage

2001-411: Is perpendicular to the line. Likewise, the distance from a point to a curve is measured by a line segment that is perpendicular to a tangent line to the curve at the nearest point on the curve. The distance from a point to a plane is measured as the length from the point along a segment that is perpendicular to the plane, meaning that it is perpendicular to all lines in the plane that pass through

2070-524: Is that they correspond to different eigenvalues. This means, in Dirac notation , that ⟨ ψ m | ψ n ⟩ = 0 {\displaystyle \langle \psi _{m}|\psi _{n}\rangle =0} if ψ m {\displaystyle \psi _{m}} and ψ n {\displaystyle \psi _{n}} correspond to different eigenvalues. This follows from

2139-400: Is the generalization of the geometric notion of perpendicularity . Whereas perpendicular is typically followed by to when relating two lines to one another (e.g., "line A is perpendicular to line B"), orthogonal is commonly used without to (e.g., "orthogonal lines A and B"). Orthogonality is also used with various meanings that are often weakly related or not related at all with

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2208-453: Is traditionally credited with proving the theorem; however, even by the 5th century BC there was nothing extant of Thales' writing, and inventions and ideas were attributed to men of wisdom such as Thales and Pythagoras by later doxographers based on hearsay and speculation. Reference to Thales was made by Proclus (5th century AD), and by Diogenes Laërtius (3rd century AD) documenting Pamphila 's (1st century AD) statement that Thales "was

2277-591: The web site of the Thyssen-Bornemisza Museum states that "Mondrian ... dedicated his entire oeuvre to the investigation of the balance between orthogonal lines and primary colours." Archived 2009-01-31 at the Wayback Machine Orthogonality in programming language design is the ability to use various language features in arbitrary combinations with consistent results. This usage was introduced by Van Wijngaarden in

2346-633: The 31st proposition in the third book of Euclid 's Elements . It is generally attributed to Thales of Miletus , but it is sometimes attributed to Pythagoras . Non si est dare primum motum esse o se del mezzo cerchio far si puote triangol sì c'un recto nonauesse. – Dante's Paradiso , Canto 13, lines 100–102 Non si est dare primum motum esse, Or if in semicircle can be made Triangle so that it have no right angle. – English translation by Longfellow Babylonian mathematicians knew this for special cases before Greek mathematicians proved it. Thales of Miletus (early 6th century BC)

2415-612: The bilinear form, the vector space may contain null vectors , non-zero self-orthogonal vectors, in which case perpendicularity is replaced with hyperbolic orthogonality . In the case of function spaces , families of functions are used to form an orthogonal basis , such as in the contexts of orthogonal polynomials , orthogonal functions , and combinatorics . In optics , polarization states are said to be orthogonal when they propagate independently of each other, as in vertical and horizontal linear polarization or right- and left-handed circular polarization . In special relativity ,

2484-461: The circle of radius OH with centre H . OP is a diameter of this circle, so the triangles connecting OP to the points T and T′ where the circles intersect are both right triangles. Thales's theorem can also be used to find the centre of a circle using an object with a right angle, such as a set square or rectangular sheet of paper larger than the circle. The angle is placed anywhere on its circumference (figure 1). The intersections of

2553-418: The design of Algol 68 : The number of independent primitive concepts has been minimized in order that the language be easy to describe, to learn, and to implement. On the other hand, these concepts have been applied “orthogonally” in order to maximize the expressive power of the language while trying to avoid deleterious superfluities. Orthogonality is a system design property which guarantees that modifying

2622-400: The design of both the transmitter and the receiver. In conventional FDM, a separate filter for each subchannel is required. When performing statistical analysis, independent variables that affect a particular dependent variable are said to be orthogonal if they are uncorrelated, since the covariance forms an inner product. In this case the same results are obtained for the effect of any of

2691-488: The diagonals is perpendicular to the opposite side. By van Aubel's theorem , if squares are constructed externally on the sides of a quadrilateral, the line segments connecting the centers of opposite squares are perpendicular and equal in length. Up to three lines in three-dimensional space can be pairwise perpendicular, as exemplified by the x, y , and z axes of a three-dimensional Cartesian coordinate system . Orthogonality In mathematics , orthogonality

2760-494: The dimensions are large, and great accuracy is not needed. The chains can be used repeatedly whenever required. If two lines ( a and b ) are both perpendicular to a third line ( c ), all of the angles formed along the third line are right angles. Therefore, in Euclidean geometry , any two lines that are both perpendicular to a third line are parallel to each other, because of the parallel postulate . Conversely, if one line

2829-402: The ellipse. The major axis of an ellipse is perpendicular to the directrix and to each latus rectum . In a parabola , the axis of symmetry is perpendicular to each of the latus rectum, the directrix, and the tangent line at the point where the axis intersects the parabola. From a point on the tangent line to a parabola's vertex, the other tangent line to the parabola is perpendicular to

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2898-428: The equality of the base angles of an isosceles triangle, ∠ OBC = ∠ OCB and ∠ OBA = ∠ OAB . Let α = ∠ BAO and β = ∠ OBC . The three internal angles of the ∆ ABC triangle are α , ( α + β ) , and β . Since the sum of the angles of a triangle is equal to 180°, we have Q.E.D. The theorem may also be proven using trigonometry : Let O = (0, 0) , A = (−1, 0) , and C = (1, 0) . Then B

2967-420: The explanatory variables and model residuals. In taxonomy , an orthogonal classification is one in which no item is a member of more than one group, that is, the classifications are mutually exclusive. In chemistry and biochemistry, an orthogonal interaction occurs when there are two pairs of substances and each substance can interact with their respective partner, but does not interact with either substance of

3036-858: The fact that Schrödinger's equation is a Sturm–Liouville equation (in Schrödinger's formulation) or that observables are given by Hermitian operators (in Heisenberg's formulation). In art, the perspective (imaginary) lines pointing to the vanishing point are referred to as "orthogonal lines". The term "orthogonal line" often has a quite different meaning in the literature of modern art criticism. Many works by painters such as Piet Mondrian and Burgoyne Diller are noted for their exclusive use of "orthogonal lines" — not, however, with reference to perspective, but rather referring to lines that are straight and exclusively horizontal or vertical, forming right angles where they intersect. For example, an essay at

3105-433: The fact that the inner product vanishes for perpendicular vectors: Both proofs are valid for horizontal and vertical lines to the extent that we can let one slope be ε {\displaystyle \varepsilon } , and take the limit that ε → 0. {\displaystyle \varepsilon \rightarrow 0.} If one slope goes to zero, the other goes to infinity. Each diameter of

3174-654: The first to inscribe in a circle a right-angle triangle". Thales was claimed to have traveled to Egypt and Babylonia , where he is supposed to have learned about geometry and astronomy and thence brought their knowledge to the Greeks, along the way inventing the concept of geometric proof and proving various geometric theorems. However, there is no direct evidence for any of these claims, and they were most likely invented speculative rationalizations. Modern scholars believe that Greek deductive geometry as found in Euclid's Elements

3243-543: The geometric sense discussed above, both as observed data (i.e., vectors) and as random variables (i.e., density functions). One econometric formalism that is alternative to the maximum likelihood framework, the Generalized Method of Moments , relies on orthogonality conditions. In particular, the Ordinary Least Squares estimator may be easily derived from an orthogonality condition between

3312-571: The graphs of the functions will be perpendicular if m 1 m 2 = − 1. {\displaystyle m_{1}m_{2}=-1.} The dot product of vectors can be also used to obtain the same result: First, shift coordinates so that the origin is situated where the lines cross. Then define two displacements along each line, r → j {\displaystyle {\vec {r}}_{j}} , for ( j = 1 , 2 ) . {\displaystyle (j=1,2).} Now, use

3381-465: The greater segment is greater than a right angle, and the angle of the less segment is less than a right angle." Dante Alighieri 's Paradiso (canto 13, lines 101–102) refers to Thales's theorem in the course of a speech. The following facts are used: the sum of the angles in a triangle is equal to 180° and the base angles of an isosceles triangle are equal. Since OA = OB = OC , △ OBA and △ OBC are isosceles triangles, and by

3450-454: The independent variables upon the dependent variable, regardless of whether one models the effects of the variables individually with simple regression or simultaneously with multiple regression . If correlation is present, the factors are not orthogonal and different results are obtained by the two methods. This usage arises from the fact that if centered by subtracting the expected value (the mean), uncorrelated variables are orthogonal in

3519-448: The instruction fields. One field identifies the registers to be operated upon and another specifies the addressing mode. An orthogonal instruction set uniquely encodes all combinations of registers and addressing modes. In telecommunications , multiple access schemes are orthogonal when an ideal receiver can completely reject arbitrarily strong unwanted signals from the desired signal using different basis functions . One such scheme

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3588-428: The left and right stereo channels in a single groove. The V-shaped groove in the vinyl has walls that are 90 degrees to each other, with variations in each wall separately encoding one of the two analogue channels that make up the stereo signal. The cartridge senses the motion of the stylus following the groove in two orthogonal directions: 45 degrees from vertical to either side. A pure horizontal motion corresponds to

3657-406: The line from that point through the parabola's focus . The orthoptic property of a parabola is that If two tangents to the parabola are perpendicular to each other, then they intersect on the directrix. Conversely, two tangents which intersect on the directrix are perpendicular. This implies that, seen from any point on its directrix, any parabola subtends a right angle. The transverse axis of

3726-479: The mathematical meanings. The word comes from the Ancient Greek ὀρθός ( orthós ), meaning "upright", and γωνία ( gōnía ), meaning "angle". The Ancient Greek ὀρθογώνιον ( orthogṓnion ) and Classical Latin orthogonium originally denoted a rectangle . Later, they came to mean a right triangle . In the 12th century, the post-classical Latin word orthogonalis came to mean

3795-428: The nearest point in the plane to the given point. Other instances include: Perpendicular regression fits a line to data points by minimizing the sum of squared perpendicular distances from the data points to the line. Other geometric curve fitting methods using perpendicular distance to measure the quality of a fit exist, as in total least squares . The concept of perpendicular distance may be generalized to In

3864-481: The other pair. For example, DNA has two orthogonal pairs: cytosine and guanine form a base-pair, and adenine and thymine form another base-pair, but other base-pair combinations are strongly disfavored. As a chemical example, tetrazine reacts with transcyclooctene and azide reacts with cyclooctyne without any cross-reaction, so these are mutually orthogonal reactions, and so, can be performed simultaneously and selectively. In organic synthesis , orthogonal protection

3933-400: The parallelogram, are both diameters of the circle and therefore have equal length, the parallelogram must be a rectangle. All angles in a rectangle are right angles. For any triangle, and, in particular, any right triangle, there is exactly one circle containing all three vertices of the triangle. ( Sketch of proof . The locus of points equidistant from two given points is a straight line that

4002-420: The point P using Thales's theorem , see the animation at right. The Pythagorean theorem can be used as the basis of methods of constructing right angles. For example, by counting links, three pieces of chain can be made with lengths in the ratio 3:4:5. These can be laid out to form a triangle, which will have a right angle opposite its longest side. This method is useful for laying out gardens and fields, where

4071-557: The point P using compass-and-straightedge construction , proceed as follows (see figure left): To prove that the PQ is perpendicular to AB, use the SSS congruence theorem for QPA' and QPB' to conclude that angles OPA' and OPB' are equal. Then use the SAS congruence theorem for triangles OPA' and OPB' to conclude that angles POA and POB are equal. To make the perpendicular to the line g at or through

4140-401: The prone to error device or method. The failure mode of an orthogonally redundant back-up device or method does not intersect with and is completely different from the failure mode of the device or method in need of redundancy to safeguard the total system against catastrophic failure. In neuroscience , a sensory map in the brain which has overlapping stimulus coding (e.g. location and quality)

4209-420: The reliability of the measurement. Orthogonal testing thus can be viewed as "cross-checking" of results, and the "cross" notion corresponds to the etymologic origin of orthogonality . Orthogonal testing is often required as a part of a new drug application . In the field of system reliability orthogonal redundancy is that form of redundancy where the form of backup device or method is completely different from

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4278-404: The same point on a circle but going through opposite endpoints of a diameter are perpendicular. This is equivalent to saying that any diameter of a circle subtends a right angle at any point on the circle, except the two endpoints of the diameter. The major and minor axes of an ellipse are perpendicular to each other and to the tangent lines to the ellipse at the points where the axes intersect

4347-456: The scope, content, and purpose of the pieces of writing are entirely unrelated. In board games such as chess which feature a grid of squares, 'orthogonal' is used to mean "in the same row/'rank' or column/'file'". This is the counterpart to squares which are "diagonally adjacent". In the ancient Chinese board game Go a player can capture the stones of an opponent by occupying all orthogonally adjacent points. Stereo vinyl records encode both

4416-406: The square of the diameter. The sum of the squared lengths of any two perpendicular chords intersecting at a given point is the same as that of any other two perpendicular chords intersecting at the same point, and is given by 8 r – 4 p (where r is the circle's radius and p is the distance from the center point to the point of intersection). Thales' theorem states that two lines both through

4485-591: The technical effect produced by a component of a system neither creates nor propagates side effects to other components of the system. Typically this is achieved through the separation of concerns and encapsulation , and it is essential for feasible and compact designs of complex systems. The emergent behavior of a system consisting of components should be controlled strictly by formal definitions of its logic and not by side effects resulting from poor integration, i.e., non-orthogonal design of modules and interfaces. Orthogonality reduces testing and development time because it

4554-407: The triangle is right, and the center of its circumcircle lies on its hypotenuse. The converse of Thales's theorem is then: the center of the circumcircle of a right triangle lies on its hypotenuse. (Equivalently, a right triangle's hypotenuse is a diameter of its circumcircle.) This proof consists of 'completing' the right triangle to form a rectangle and noticing that the center of that rectangle

4623-401: The two lines intersect at a right angle. Explicitly, a first line is perpendicular to a second line if (1) the two lines meet; and (2) at the point of intersection the straight angle on one side of the first line is cut by the second line into two congruent angles . Perpendicularity can be shown to be symmetric , meaning if a first line is perpendicular to a second line, then the second line

4692-452: The two-dimensional plane, right angles can be formed by two intersected lines if the product of their slopes equals −1. Thus for two linear functions y 1 ( x ) = m 1 x + b 1 {\displaystyle y_{1}(x)=m_{1}x+b_{1}} and y 2 ( x ) = m 2 x + b 2 {\displaystyle y_{2}(x)=m_{2}x+b_{2}} ,

4761-458: Was not developed until the 4th century BC, and any geometric knowledge Thales may have had would have been observational. The theorem appears in Book III of Euclid's Elements ( c.  300 BC ) as proposition 31: "In a circle the angle in the semicircle is right, that in a greater segment less than a right angle, and that in a less segment greater than a right angle; further the angle of

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