A cathedral is erected on two hills, the side circular arcs whose centers are the hill ends, and so that the right (and left) side arcs are orthogonal. Show that the tip of the spire is directly above where the hills meet.

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## Solution

Y and Z are inverses of X in the two side circles (property of orthogonal circles) and a little algebra shows that the powers of X for the two circles are equal, so X is on their radical axis, which is orthogonal to their line of centres.

## One reply on “Cathedral on the hills”

The solution should have shown what Matthew Arcus said to go with the diagram:

“Y and Z are inverses of X in the two side circles (property of orthogonal circles) and a little algebra shows that the powers of X for the two circles are equal, so X is on their radical axis, which is orthogonal to their line of centres:”