Naming convention for circles in geometry
In geometry, do problems usually refer to the circle as a name, such as circle $A$, then name the center? Or do they name the circle the same as the center, such as $A$? The reason I ask is because I was doing a geometry problem where I think they were saying the latter and wanted to know if this ever does happen.
Here is the question:
$\endgroup$ 1Circles A and B are tangent and have radii 1 and 2, respectively. A tangent to circle A from the point B intersects circle A at C. D is chosen on circle B so that AC is parallel to BD and the two segments BC and AD do not intersect. Segment AD intersects circle A at E. The line through B and E intersects circle A through another point F. Find EF.
2 Answers
$\begingroup$- A circle with center $A$ is often denoted $\bigcirc A$.
As in the quoted problem, an author may opt to "write-out" a symbol that may not be available for type-setting; for instance, "circle $A$" for "$\bigcirc A$". In that case, it's incumbent on the author to make clear whether a particular instance of "$A$" in the text refers to the circle itself or its center-point. (The quoted problem seems consistent at referring to "circle $A$".)
Sometimes, because the center isn't relevant to the statement of a problem, or because it's unknown, or because there may be multiple circles with the same center, or for whatever reason, a circle itself may be given a letter name. Convention favors a Greek letter, in particular
gamma, either upper-case $\Gamma$ or lower-case $\gamma$. The "$\bigcirc$" may-or-may-not appear in this notation; usually not.Occasionally, the circumcircle of $\triangle ABC$ may be named $\bigcirc ABC$. I have also seen $(ABC)$, although I personally would resist yet-another overloading of parentheses.
This is not unlike naming conventions for lines. When it's helpful to describe a line as passing through two known points $A$ and $B$, we denote it $\overleftrightarrow{AB}$. (In the absence of fancy type-setting, we might see "line $AB$" or "segment $AB$" for "$\overleftrightarrow{AB}$" or "$\overline{AB}$".) If we don't know or care about specific points on the line, we can choose assign a letter name, usually lower-case, such as $\ell$ (again, with the "$\overleftrightarrow{}$" usually suppressed).
$\endgroup$ $\begingroup$No, it is usual to refer to it as the circle $A$ with center $a_0$ (or whatever name you prefer to use) with coordinates $(x_0,y_0)$.
It is not normal to denote a circle and its center by the same name.
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