triangle ABC is a triangle with no equal sides. A new triangle congruent to triangle ABC is to be placed on the plane such that the new triangle shares exactly two vertices with triangle ABC. At how many different locations can the new triangle be placed?
The answer is 9!
If the new triangle shares two vertices with triangle abc, it shares the side between those two vertices. Given an edge of triangle ABC to be shared, there is only one edge of the new triangle to share it: the one with the same length. There are four ways to choose how an edge XY of the new triangle share an edge, say , AB of the original: X could fall on A or on B (causing Y to fall on B or A, respectively), and the third vertex could be on either of the two sides of line AB, giving 2*2=4 possibilities. One of these possibilities is forbidden, though, because it would have all three vertices of the new triangle sharing with triangle ABC. Thus, there are three legal ways that a given edge of triangle ABC can be shared.
Since there are three ways that the new triangle can be placed to share a given edge and three possible edges to share, our answer is 3*3=9.