How can you check if a point is the circumcenter of a triangle?

Check if the point is at the same distance from all the vertices of the triangle.

Check if the point is the point of intersection for the angle bisectors of the triangle.

Check if any of the perpendicular bisectors of the triangle passes through the point.

Check if the point is the center of the largest circle that can be drawn inside the triangle.

Question

How can you check if a point is the circumcenter of a triangle?

Check if the point is at the same distance from all the vertices of the triangle.
		
Check if the point is the point of intersection for the angle bisectors of the triangle.
		
Check if any of the perpendicular bisectors of the triangle passes through the point.
		
Check if the point is the center of the largest circle that can be drawn inside the triangle.

accessdenied · Answer

the circumcenter is the center of a circle that can be circumscribed about the triangle, so the radius to each of the vertices would be equal (A).  Also, it is the point of concurrency for all three perpendicular bisectors of the triangle, so it seems like (C) is correct as well...

anonymous · Answer

i was gonna choose c any way just wanted to check thanks

accessdenied · Answer

..but (C) is less specific -- we could have any points on that perpendicular bisector inside the triangle -- (A) has to be correct!

anonymous · Answer

only C

anonymous · Answer

http://www.mathopenref.com/trianglecircumcenter.html

accessdenied · Answer

" The circumcenter is also the center of the triangle's circumcircle - the circle that passes through all three of the triangle's vertices."
the center of a circle that touches all three points

accessdenied · Answer

which means the radius is the same between all three points if it is the center of that circle

anonymous · Answer

I guess A is also  right

accessdenied · Answer

the problem I have with C is that it says 'any of the perpendicular bisectors', except that doesn't necessarily mean we check all of them...

anonymous · Answer

access denied could you then please give me example proving a is answer understand the hole the radius is the same around circle but how does it apply to triangle

accessdenied · Answer

the circle we are talking about touches all three vertices of the triangle.

i'll try to draw it, though it may not turn out as nicely as I hope

|dw:1330815783048:dw|
^ this would be the circumcenter (the point of concurrency for the perpendicular bisectors), and as we can see, the circle around it is equidistant from the point (at least, sorta in my drawing)

basically, the radius of the circle is also the distance from the circumcenter to each vertex (since the circle contains those points; they are points on the circle)

accessdenied · Answer

|dw:1330816088607:dw|
^ that might help visualize what I'm trying to explain...

anonymous · Answer

oh thanks so much u helped alot :)

accessdenied · Answer

No problem!  Glad to help! : D