Let \(C_1\) and \(C_2\) be circles that intersect at right angles, having \(P_1\) and \(P_2\) as their centres. Then the circle with diameter \( \vec{P_1 P_2} \) has a quarter of the area of the sum of the areas of \( C_1 \) and \( C_2 \).

Spoiler 

@christianp Like so. Maybe it's the tangents that do that, but I'd argue that it's fine in the limit.
mathstodon.xyz/media/IkoU5xjqj

Spoiler 

@christianp Take any two points on the circumference of a circle, draw the tangents at them, draw a circle with the tangent intersection as its centre. Well-known construction for some examples of hyperbolic geometries.

CC: @icecolbeveridge

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