https://www.google.com/search?q=family of circles geometry "...

HTH :-)

My daughter is a mechanical engineer and secondary school maths teacher. She's just shown me examples in Spanish and we've found this translation.

Pencil of circles

The Apollonian circles, two orthogonal pencils of circles
Any two circles in the plane have a common radical axis, which is the line consisting of all the points that have the same power with respect to the two circles. A pencil of circles (or coaxial system) is the set of all circles in the plane with the same radical axis. To be inclusive, concentric circles are said to have the line at infinity as a radical axis.

https://en.wikipedia.org/wiki/Pencil_(geometry)

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Note added at 2 hrs (2024-05-07 16:56:48 GMT)
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On the same Wikipedia page there's a reference for haz de circunferencias

La circunferencia de Apolonio es un famoso problema acerca de lugares geométricos: dados dos puntos A y B, se trata de determinar el lugar geométrico de los puntos del plano P que cumplen: PA/PB = r, siendo r una constante.

I clicked on the link and then clicked on the English version and found:

In geometry, Apollonian circles are two families (pencils) of circles such that every circle in the first family intersects every circle in the second family orthogonally, and vice versa. These circles form the basis for bipolar coordinates. They were discovered by Apollonius of Perga, a renowned Greek geometer.

https://en.wikipedia.org/wiki/Apollonian_circles


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A pencil of circles is another name for specific families of circles who all share certain characteristics within the family.

An elliptical pencil of circles is the family of all circles that go through two given points. This is why another name for them is a two-point pencil. These circles are all intersecting, but never anywhere but these two points.

A hyperbolic pencil of circles, also called Apollonian circles, is the family of all circles that are orthogonal to the set of circles that go through 2 given points. Thus, the elliptical pencil of circles always has a corresponding hyperbolic pencil of circles that are orthogonal to every circle. None of the circles in the hyperbolic pencil intersect with each other, thus they are also called the zero-point pencil.

The third kind of pencil is the parabolic pencil. This is the family of circles which all have one common point, and thus are all tangent to each other, either internally or externally. Also, the orthogonal set of circles to a parabolic pencil is another parabolic pencil.

https://sites.math.washington.edu//~king/coursedir/m445w04/a...

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A family of circles that satisfy #2 and #6 is said to form a pencil of circles. The circles are said to be coaxal (sometimes coaxial), because all of them have the same axis of symmetry and any pair has the same radical axis. This property is obviously shared by a family of circles through a given point, say, A and tangent at A to each other. Perhaps, even more interestingly, the same holds for the family of circles that pass through two given points, A and B: their centers lie on the perpendicular bisector of AB; in addition, the radical axis of any pair passes through their common points: A and B. Since any pairs of circles that have the same radical axis are bound to have their centers collinear, sharing the radical axis by any pair of circles in the family is the defining property of the circles.

https://www.cut-the-knot.org/Curriculum/Geometry/CoaxalCircl...

http://users.math.uoc.gr/~pamfilos/eGallery/problems/CircleB...

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Note added at 9 hrs (2024-05-08 00:47:58 GMT)
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https://www.google.com/search?q=circle bundles&sca_esv=fa48f...

In geometrical mathematics a circumference is the boundary of a specific geometric figure, particularly a circle, or it's the length of a closed geometric curve. (thefreedictionary.com)
"The radical axis of 2 non-concentric circles is the set of points whose power with respect to the circles is equal."
https://en.wikipedia.org/wiki/Radical_axis
"A beam is a structural element which resists loads primarily. It's applied laterally across the axis of a beam.
Traditionally beams are descriptions of building or civil engineering structural elements where the beams are horizontal and carry or support vertical loads."
https://en.wikipedia.org/wiki/Beam_(structure)

Efectivamente, el original se refiere al conjunto de círculos que comparte un mismo eje radical.

Some refs:

Coaxial circles
https://dlmf.nist.gov/search/search?q=coaxial circles

Coaxial system of circles
https://www.toppr.com/ask/question/a-system-of-circles-is-sa...

Two orthogonal pencils of coaxial circles. Dashed circles are centered on the y axis and go through the points −1 and 1. Dotted circles are centered on the x axis and intersect dashed circles at the right angle. 
https://www.researchgate.net/figure/Two-orthogonal-pencils-o...

Circles (F2AA'), (F2BB'), (F2CC') are coaxial
https://artofproblemsolving.com/community/c2735h1454926_ferm...

If a set of circles have the same radical axes, then we say they are coaxial. A collection of such circles is called a pencil of coaxial circles.
https://www.google.com/url?sa=t&source=web&rct=j&opi=8997844...