[Maths-Education] concentric and congruent circles

[Kate Mackrell]

I'm in the middle of quite an interesting experience regarding the  
definition of these mathematical terms, which I thought might be worth  
sharing.

I'm developing some files for an Asian educational software company,  
and in the course of this development have been discovering that many  
terms have slightly different definitions in different places.  One  
example is "adjacent angles".  To me, these are angles on a straight  
line, that  add up to 180.  To this company, these are angles with one  
common arm and a common vertex and no specific sum.  Another example  
is of course the trapezoid/trapezium - does it have only one or at  
least one pair of parallel sides?  I've been quite happy to go along  
with the the company's terminology (I rather prefer their definition  
of adjacent angles), but we've now hit a sticking point on concentric  
and congruent circles.

To me (and Wolfram, and lots of other authoritative online  
dictionaries of mathematics), two circles are concentric if they have  
the same centre.  The radii of the circles are immaterial - hence two  
congruent circles may or may not be concentric.  To the people I'm  
working for, concentric circles necessarily have different radii.

I've been quite puzzled by this, but am finally beginning to  
understand that it might be about identity: if two circles precisely  
overlap (i.e. are congruent and concentric), can we say that two  
distinct circles exist?  Experience with dynamic geometry software  
gives me the automatic answer of "of course" - I can drag two separate  
objects about the screen, make them coincide, and then drag them apart  
again.  I can also construct the two circles to be congruent - but to  
have quite different properties.  The centre of one circle could be  
free to move anywhere, while the centre of the other circle might be  
constrained to a line.  The radius of one circle might change by  
dragging on the circle - and this could determine the radius of the  
other circle, but not vice-versa.  I've never thought to question the  
Euclidean definition of a circle until now - but, in a dynamic  
geometry environment, it seems to me that it's simply not sufficient,  
as it does not give me any way of distinguishing between two congruent  
circles that happen to coincide.

On the other hand, the company wants to still call two coinciding  
circles "congruent", which implies that there are still two circles in  
a relationship with each other - are they just being inconsistent?

I also find the question interesting of when to consider it legitimate  
for a mathematical term to have a flexibly  determined meaning (such  
as trapezoid/trapezium or adjacent angles) and when to not consider it  
legitimate.  "Concurrent" has not got any of the mathematical  
inevitability of "-" x "-" = "+".  I shouldn't be concerned about its  
definition being changed.  But I find that I am very unwilling to  
change.

Thoughts, anyone?

Kate Mackrell