Given an equilateral triangle, the counterclockwise rotation by 120 ° around the center of the triangle " acts " on the set of vertices of the triangle by mapping every vertex to another one.

Given a general triangle ABC, the following conditions would need to be fulfilled for the case to be ambiguous:

* Given a triangle OAB in which O is the center of a circle k, and points A ' and B ' inverses of A and B with respect to k, then

Given a fundamental domain for the group action ( for the full, orientation-reversing symmetry group, a ( 2, 3, 7 ) triangle ), the reflection domains ( images of this domain under the group ) give a tiling of the quartic such that the automorphism group of the tiling equals the automorphism group of the surface – reflections in the lines of the tiling correspond to the reflections in the group ( reflections in the lines of a given fundamental triangle give a set of 3 generating reflections ).

Given a triangle ABC, and a triangulation T of the triangle.

Given a counterfactual conditional, e. g., ' If there had been a circle on the blackboard then there would have been a triangle ', and the subsequent information ' in fact there was no triangle ', participants make the modus tollens inference ' there was no circle ' more often than they do from an indicative conditional ( Byrne and Tasso, 1999 ).

Given a unit sphere, a " triangle " on the surface of the sphere is defined by the great circles connecting three points u, v, and w on the sphere.

Given a point inside a triangle it is also desirable to obtain the barycentric coordinates, and at this point.

Given a triangle ABC and its three Malfatti circles, let D, E, and F be the points where two of the circles touch each other, opposite vertices A, B, and C respectively.

Given a triangle ABC, and a transversal line that crosses BC, AC and AB at points D, E and F respectively, with D, E, and F distinct from A, B and C, then

Given An equilateral triangle inscribed on a circle and a point on the circle.

The chords are of considerable historical importance because, along with the sides of the triangle and tetragon ( square ), they enable the generation of a table of half chords ( effectively sine values ) Given the unit circle with sector ABO subtending an arc of, we may write − a relationship expressed in words by Copernicus:

Given any natural numbers l, m, n > 1 exactly one of the classical two-dimensional geometries ( Euclidean, spherical, or hyperbolic ) admits a triangle with the angles ( π / l, π / m, π / n ), and the space is tiled by reflections of the triangle.

Given any Euclidean triangle ABC and an arbitrary point P let d ( P ) = PA + PB + PC.

Given any triangle ABC, and any point M on BC, construct the incircle and circumcircle of the triangle.

The has a crossing point at the origin if there is a triangle with sides a, b and c. Given the previous conditions, this means that the curve crosses the origin if and only if b < a + c.

: Given three points A, B and C on a circle with center O, the angle AOC is twice as large as the angle ABC.

Given Carson's undisputed status as the dean of late-night television, NBC heard his request as an ultimatum, fearing he might use the issue as grounds to defect to either ABC or CBS.

The choir have recorded three CDs-Christmas at St James ( 2003 ), No Ordinary Sunday ( 2004 ) and Any Given Sunday ( 2007 )-and broadcasts regularly on ABC Radio, both in their own right as well as with leading ensembles such as Australian Baroque Brass.

Given Nexstar's termination of the Fox affiliations of several of its stations during the past two years, it is ironic that KLRT is one of two Fox affiliates that are being picked up by either Nexstar or Mission ; the other is WJKT in Jackson, Tennessee ( although that station is controlled by and based out of Memphis, Tennessee's ABC affiliate WPTY-TV, which is also being sold to Nexstar ).

Given two groups G and H and a group homomorphism f: G → H, let K be a normal subgroup in G and φ the natural surjective homomorphism G → G / K ( where G / K is a quotient group ).

Given a groupoid in the category-theoretic sense, let G be the disjoint union of all of the sets G ( x, y ) ( i. e. the sets of morphisms from x to y ).

Given a topological space X, let G < sub > 0 </ sub > be the set X.

Given a point x in a topological space, let N < sub > x </ sub > denote the set of all neighbourhoods containing x.

Given two points P and Q on C, let s ( P, Q ) be the arc length of the portion of the curve between P and Q and let d ( P, Q ) denote the length of the line segment from P to Q.

Given a linearly recursive sequence, let C be the transpose of the companion matrix of its characteristic polynomial, that is

Given a linear homogeneous recurrence relation with constant coefficients of order d, let p ( t ) be the characteristic polynomial ( also " auxiliary polynomial ")

Given that the police had no evidence against " Thomas ", either at Leatherslade Farm or connection with either of the two gangs, Butler was prepared to let him go.

Given a concrete category ( C, U ) and a cardinal number N, let U < sup > N </ sup > be the functor C → Set determined by U < sup > N </ sup >( c ) = ( U ( c ))< sup > N </ sup >.

Given a unit vector n in R < sup > 3 </ sup > and an angle φ, let R ( φ, n ) represent a counterclockwise rotation about the axis through n ( with orientation determined by n ).

Given a subset V of P < sup > n </ sup >, let I ( V ) be the ideal generated by all homogeneous polynomials vanishing on V. For any projective algebraic set V, the coordinate ring of V is the quotient of the polynomial ring by this ideal.

Given a homogeneous prime ideal P of, let X be a subset of P < sup > n </ sup >( k ) consisting of all roots of polynomials in P .< ref > The definition makes sense since if and only if for any nonzero λ in k .</ ref > Here we show X admits a structure of variety by showing locally it is an affine variety.

Given an integral domain, let be an element of, the polynomial ring with coefficients in.

) Given a section σ of E let the corresponding equivariant map be ψ ( σ ).

Given an integer n > 1, let H be any subgroup of the multiplicative group

" Given the massive undertaking in creating 11 volumes over 50 years, errors and incompleteness were inevitable by Durant's own reckoning ; but no other historical survey matches let alone exceeds the breadth and depth of his project, he claimed.

Given the military-industrial requirements of the Cold War, the authors of the then secret National Security Council Report 68 ( NSC-68 ) proposed the US government undertake a massive permanent national economic expansion that would let it “ siphon off ” a part of the economic activity produced to support an ongoing military buildup to contain the Soviet Union.

Let E be a rank k vector bundle over a smooth manifold M and let ∇ be a connection on E. Given a piecewise smooth loop γ: → M based at x in M, the connection defines a parallel transport map P < sub > γ </ sub >: E < sub > x </ sub > → E < sub > x </ sub >.

Given a compact subset K of X and an open subset U of Y, let V ( K, U ) denote the set of all functions such that Then the collection of all such V ( K, U ) is a subbase for the compact-open topology on C ( X, Y ).

Given a set of primitive types, let

Given a semiautomaton, let T < sub > a </ sub >: X → X, for a ∈ Σ, denote the transformation of X defined by T < sub > a </ sub >( x ) = T ( a, x ).

: Given such a tangent vector v, let f be the curve given in the x < sup > i </ sup > coordinate system by.