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 ABC, let the lines AO, BO and CO be drawn from the vertices to a common point O to meet opposite sides at D, E and F respectively.

* 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 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 a direct product of groups, then H is a transversal for the cosets of K.

Given points P < sub > 0 </ sub > and P < sub > 1 </ sub >, a linear Bézier curve is simply a straight line between those two points.

Given this basic definition, further criteria are needed to establish a dividing line between postlexical clitics and morphological affixes, since both are characterized by a lack of prosodic autonomy.

: Given two points, determine the azimuth and length of the line ( straight line, arc or geodesic ) that connects them.

Given a function f of a real variable x and an interval of the real line, the definite integral

# Given any two distinct points, there is exactly one line incident with both of them.

Given ailing William III's reluctance to remarry, inclusion of Sophia in the line of succession was becoming more likely.

Given India's historic position that the original intent of the line was to separate the two nations by the highest mountains in the world, in these locations India extended its forward posts northward to the ridges, regarding this move as compliant with the original border proposal, although the Simla Convention did not explicitly state this intention.

Given the consensus that " the most archaic layer of the biography, that of the stories of the kussās popular story-tellers " ( Sīra, EI² ), this may prove a fruitful line of inquiry.

Given any curve C and a point P on it, there is a unique circle or line which most closely approximates the curve near P, the osculating circle at P. The curvature of C at P is then defined to be the curvature of that circle or line.

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 set of points in Euclidean space, the first principal component corresponds to a line that passes through the multidimensional mean and minimizes the sum of squares of the distances of the points from the line.

Hypothesis: Given AD is a straight line, and AB, BC, and CD are all equal length,

* Given two points, to draw the line connecting them.

Given that the two endpoints of the line segment are specified at positions ( x1, y1 ) and ( x2, y2 ).

Given the two red points, the blue line is the linear interpolant between the points, and the value y at x may be found by linear interpolation.

Given the death of Princess Charlotte without surviving children, should the Ord or Lowe links be substantiated, the line descended through them would join a large number of claimed surviving descendants of King George IV.

Given a point in the plane, draw a straight line connecting it with the north pole on the sphere.

" Given that the parched region of the Gran Chaco ( largely uninhabited ) had been under dispute between Bolivia and Paraguay ever since the creation of both republics, each proceeded to establish a line of small garrisons ( fortines ), simply to establish a national presence and press their claims.

# Given a point p < sub > 1 </ sub > and a line l < sub > 1 </ sub >, there is a unique fold perpendicular to l < sub > 1 </ sub > that passes through point p < sub > 1 </ sub >.

# Given two points p < sub > 1 </ sub > and p < sub > 2 </ sub > and a line l < sub > 1 </ sub >, there is a fold that places p < sub > 1 </ sub > onto l < sub > 1 </ sub > and passes through p < sub > 2 </ sub >.

Given a point p < sub > 1 </ sub > and a line l < sub > 1 </ sub >, there is a unique fold perpendicular to l < sub > 1 </ sub > that passes through point p < sub > 1 </ sub >.

Given two points p < sub > 1 </ sub > and p < sub > 2 </ sub > and a line l < sub > 1 </ sub >, there is a fold that places p < sub > 1 </ sub > onto l < sub > 1 </ sub > and passes through p < sub > 2 </ sub >.