Compass Geometry

5. Construction PC to invert a point in a circle

Let there be given a circle C with center O and point A on its circumference, and a point P either outside the circle C or inside but more than half the radius away from O. Here is a construction to invert P in the circle C to get P' = P>C. Draw the circle with center P and radius PO. Then this new circle meets the circle C at two points, Q and Q'. Draw the two circles with centers Q and Q' and radii QO and QO', respectively. Besides O, they will meet at another point P'.

This point P' will be the image of the point P inverted in the circle C. The usual definition of the inverted point P' is that it is situated on the line OP from the center of the circle to P so that OP:OA = OA:OP'. From the symmetry of the construction, P' is clearly on that line. To see that the proportion is valid, note that the two triangles QOP and P'OQ are each isosceles, and they have the same base angles, so they're similar. The proportion immediately follows.

Invert P in a circle

Now, this construction doesn't work if P is within half the radius of O since the two circles won't intersect to produce the points Q and Q', so some other construction is needed to invert such P. But if P is at least a quarter of the radius away from O, then we will find the point R twice as far from O as P is, then invert R in the circle C to get R' = R>C by the method described above, and then P' will be the point twice as far from O as R' is. This requires a doubling construction 2OP.

Double OP

The construction 2OP can be effected as follows. Draw the circles with centers O and P and radius OP. Let them intersect at Q and Q'. Draw the circle with center Q and radius QQ'. It will intersect the circle with center P at the points Q' and R.

Now we can invert any point P in a circle so long as it is at least a quarter of a radius away from the center of the circle.
When P is very close to O, all we have to do is repeatedly double the distance from O until we've got a point at least half the radius from O, invert that point, and double the resulting distance the same number of times to get P'.

Invert P in a circle

Bisecting and multisecting lines

Before going on, we can use inversion in circles to effect a construction to bisect a line segment, or more generally, cut off one n-th of a line segment. To bisect a line segment OP, use construction 2OP to double it to OR, then construction PC to invert R in the circle with center O and radius OP to get the midpoint M of OP.

A midpoint construction

We can trisect OP by doubling OP to OR then doubling PR to PS, so that OS is triple OP, then inverting S in the circle with center O and radius OP to get S'. Then OS' will be one-third of OP

More generally, we can cut off an n-th of OP by first extending OP by a factor of n (which is easily seen as a variant of the doubling construction), then inverting in that same circle.

Next part: 6. Find the center of a circle

April, 1998; March, 2002.
David E. Joyce
Department of Mathematics and Computer Science
Clark University
Worcester, MA 01610

Email: djoyce@clarku.edu
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