The Ambiguous Case of SSA

The graph above shows why we can sometimes have no possible triangles sometimes one and sometimes two. Adjust the sliders for

and

. You can also drag around point

, but it will always be on the circle centered at

. If we are given two sides of a triangle and an angle that is not between them (SSA): Method 1: Check both Angles and Reject if Necessary

Method 2: Check

, the altitude from

to the opposite side Case I:

is acute Suppose we let

and

have fixed values and we just adjust the length of

. For example, let

and

. If we turn on the circle where

must lie, we can see that changing the length of

will make the circle intersect the opposite side once, twice, or not at all. Note that since

, we can express the altitude as: