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 a point and a circle, to draw either tangent.

Given a sphere of unit radius, place its center at the origin of the complex plane, oriented so that the equator on the sphere coincides with the unit circle in the plane, and the north pole is " above " the plane.

Given an orientable Haken manifold M, by definition it contains an orientable, incompressible surface S. Take the regular neighborhood of S and delete its interior from M. In effect, we've cut M along the surface S. ( This is analogous, in one less dimension, to cutting a surface along a circle or arc.

Given the principal stress state, we can use Mohr ’ s circle to solve for the maximum shear stresses our material will experience and conclude that the material will fail if:

De Tactionibus embraced the following general problem: Given three things ( points, straight lines, or circles ) in position, describe a circle passing through the given points and touching the given straight lines or circles.

: 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 a circle k, with a center O, and a point P outside of the circle, we want to construct the ( red ) tangent ( s ) to k that pass through P. Suppose the ( as yet unknown ) tangent t touches the circle in the point T. From symmetry, it is clear that the radius OT is orthogonal to the tangent.

* 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 the size of the two outer large circles, what is the size of the small circle between them?

Given a circle whose center is point O, choose three points V, C, and D on the circle.

Given a circle whose center is point O, choose three points V, C, and D on the circle.

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 the counterfactual conditional and the subsequent information ' in fact there was a circle ', participants make the modus ponens inference as often as they do from an indicative conditional.

Given three mutually tangent circles (< font color =" black "> black </ font >), what radius can a fourth tangent circle have?

* Squaring the circle: Given any circle drawing a square having the same area.

Given a theological lead, I asked what he thinks about those who find a religious significance to his plays.

Euclid poses the problem: " Given two numbers not prime to one another, to find their greatest common measure ".

Given more time to prepare for trial, Kidd likely would have been able to find the deposition Palmer gave when he was captured in Rhode Island two years earlier.

" Given that there's a gulf of 100, 000, 000 light years, he checks his diary to find out what he had been doing that day, and finds out that it was something he wished to hide.

Given a specific task to solve, and a class of functions, learning means using a set of observations to find which solves the task in some optimal sense.

Given printing practices at the time ( which included type-setting from dictation ), no two editions turned out to be identical, and it is relatively rare to find even two copies that are exactly the same.

* Sentence breaking ( also known as sentence boundary disambiguation ): Given a chunk of text, find the sentence boundaries.

Given a query, the Prolog engine attempts to find a resolution refutation of the negated query.

Given a list of cities and their pairwise distances, the task is to find the shortest possible route that visits each city exactly once and returns to the origin city.

Informally it solves the following problem: Given an integer N, find its prime factors.

The quantum circuits used for this algorithm are custom designed for each choice of N and the random a used in f ( x ) = a < sup > x </ sup > mod N. Given N, find Q = 2 < sup > q </ sup > such that < math > N ^ 2

Given Saint Jerome's comment that, " The whole world groaned in astonishment to find itself Arian ", it appears to have failed.

Given the alternative of holding cash, and thus earning 0 %, rather than lending it out, profit-seeking lenders will not lend below 0 %, as that will guarantee a loss, and a bank offering a negative deposit rate will find few takers, as savers will instead hold cash.

* Given n points in the plane, find the two with the smallest distance from each other.

* Convex hull: Given a set of points, find the smallest convex polyhedron / polygon containing all the points.

* Closest pair of points: Given a set of points, find the two with the smallest distance from each other.

Given the large numbers involved and the nature of the requirements, they argued, it was unlikely to find any members of Al Qaeda.

Given a function, the goal of the attack is to find two different inputs such that.

Given a database or a set of rules, the computer tries to find a solution matching all the desired properties.

Given this find, the UN Inspectors have requested access to the Al-Aziziyah weapons range to verify that all 155 R-400 bombs can be accounted for and proven destroyed.

Given a square matrix A, we want to find a polynomial whose roots are precisely the eigenvalues of A.

The specific problems are: Given two, three or four points on a straight line, find another point on it such that its distances from the given points satisfy the condition that the square on one or the rectangle contained by two has a given ratio either ( 1 ) to the square on the remaining one or the rectangle contained by the remaining two or ( 2 ) to the rectangle contained by the remaining one and another given straight line.

Given a bin size and a list of items with sizes to pack, find an integer and a-partition of such that for all.

; Inverse problem of Galois theory: Given a group G, find an extension of the rational number or other field with G as Galois group.

Given a morphism f: B → A the associated natural transformation is denoted Hom ( f ,–).

: Given any set A and any set B, if for every set C, C is a member of A if and only if C is a member of B, then A is equal to B.

: Given any set A and any set B, if A is a nonempty set ( that is, if there exists a member C of A ), then if A and B have precisely the same members, then they are equal.

: Given any set A and any set B, there is a set C such that, given any set D, D is a member of C if and only if D is equal to A or D is equal to B.

: Given any set A, there is a set B such that, given any set x, x is a member of B if and only if x is a member of A and φ holds for x.

: Given any set A, there is a set such that, given any set B, B is a member of if and only if B is a subset of A.

: Given any set A, there is a set B such that, for any element c, c is a member of B if and only if there is a set D such that c is a member of D and D is a member of A.

Given a core geometry, the B field needed for a given force can be calculated from ( 2 ); if it comes out to much more than 1. 6 T, a larger core must be used.

Given two candidates, A and B, A is better than B on a constraint if A incurs fewer violations than B.

With regard to African Americans, Gilman wrote in the American Journal of Sociology: “ The problem, is this: Given: in the same country, Race A, progressed in social evolution, say, to Status 10 ; and Race B, progressed in social evolution, say, to Status 4..