# Why are hyperbolic circles in the upper half plane also Euclidean circles?

In a few places I've seen talking about the upper half plane as a model of hyperbolic geometry, it's mentioned offhand that circles (in the sense of a set of points equidistant from a given point under a given metric) in the Euclidean and hyperbolic senses coincide - but that the centers are different! Is there a good way to show this?

I've tried using the hyperbolic metric by brute computation, and tried using some properties of Mobius transformations, but am stuck.