Thu 29 Jun 2006

A Geometry Puzzle

For a change in pace, an old geometry problem I came across again recently.

Suppose you are given two circles in a plane. Their centres are marked. You want to find a point P such that when you draw the four tangent lines from P to the two circles (two to each circle), the angles between the pair of tangent lines for each circle are equal. The following diagram may help here. Find the point P such that the two marked angles are equal.

If you know where the centres of the two circles are (and I'll give you that), you can solve this with only a straight-edge (not a measuring ruler; no measuring required). So you might have lost half of your straight-edge + compass construction kit since high school or whenever you last did these, but you don't need the compass here anyway (I gave you the circles, you don't need to draw them).

[Update: This is rubbish. You do need a compass to solve this problem. At least in the solution I have in mind. There is a need to construct a perpendicular line at one point and that cannot be done with a straight edge alone.]

1. For bonus points, find all the possible points P.

2. For even more bonus points, introduce a third circle into the picture (not on a the straight line connecting the centres of the first two circles). Now there is a unique point P that satisfies the condition (the angles between all three tangent pairs must be equal). How can you find it?

Topics: mathematics/puzzles