Where is the hiker? A hiker can see two church spires. One is straight in front of him. The other is directly to the left. He has a map on which the two churches are marked but no idea of the direction in which he is facing (whether it is North, South, etc.). What can he tell about where he is on the map?
The problem is from Sawyer's "Mathematician's Delight".
 A: From the given information, the hiker is standing at the vertex of a 90-degree angle between the churches. With no cues that would identify either church, Thales' theorem states that the hiker may be anywhere along a circle whose diameter is a line between the two churches, because at any point along the circle, the hiker's location forms a right triangle with the two churches, with the circle's diameter forming the hypotenuse:

However, there is one assumption that should be reasonably safe to make and that narrows the possibilities considerably: that the hiker can judge relative distance. If the hiker can say, at the very least, "I'm roughly the same distance from both churches", or "the church I am facing is noticeably closer/further then the one to my left", then that eliminates many of the possibilities, creating two arcs along the original circle that are most likely. For instance, if the hiker could say "the two churches are more or less the same distance from me", the possibilities might look like this:

The more accurately the hiker can gauge this distance (equivalently, the more confident he is in a relative distance reckoning like "the one in front of me is half the distance as the the one to my left"), the shorter the arcs become, ideally approaching two points if the measurement can be taken with exact precision (say a laser rangefinder). 
However, it is impossible to determine at which of those two points (or along which of those two arcs) the hiker is standing without further information.
