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Orthogonal Circle

Age 16 to 18
Challenge Level

It is known that given any three non intersecting circles in the plane there is another circle or straight line that cuts the three given circles at right angles. (The circle or straight line is said to be orthogonal to the 3 original circles.)

Given three circles with centres $(0, 0)$, $(3, 0)$ and $(9, 2)$ and radii $5$, $4$ and $6$ respectively find the centre and radius of the circle that cuts the three given circles at right angles. Draw the circles to check that the circle you have found appears to be orthogonal to the others.

What happens in the case of three circles with centres at $(0, 0)$, $(3, 3)$ and $(8, 8)$ and radii $1$, $2$ and $3$ respectively?

Given three circles, how can you tell without calculating which of the two cases applies, an orthogonal circle or an orthogonal straight line?