Distance to epicenter vs distance to focus I am not really a "budding seismologist." I am a chemistry/physics teacher teaching an Earth science class (low academic level) in a small-town high school. (My college coursework in earth sciences = 0) I understand the concept of triangulation to determine an earthquake's origin, and I realize there are some subtleties in wave analysis such as pP and sP waves. What I don't understand is how the triangulation determines the epicenter instead  of the focus. Wouldn't the S-P lag times for 3 stations give an intersection point at the focus since that is the actual point of origin for the waves?
This is a good question, but it really has a simple answer. How can you determine the depth of an object when you are not even calculating it? Triangulation uses circles, 2-D shapes, so when finding the solution to the equations of circles that solve all 3 circles, you only find X and Y. Z isn't even being solved for.   

Triangulation is not precise. In practice, we end up using much more than 3 stations and statistical methods to find the precise epicenter (and other methods to find the focus).