What's the most efficient way to detect triangle-triangle intersections?

One way is to check if two sides of triangle A intersect with any side of triangle B, and then check all six possibilities of a point of A inside B or a point of B inside A.

For a point inside a triangle see for example: Point in triangle test.

When we test collisions on polygons we also have a surrounding rectangle for our polygons. So we first test for rectangle collisions and if there is a hit we proceed with polygon collision detection.

Python implementation for line intersection and point in triangle test, with a little modification.

def line_intersect2(v1,v2,v3,v4):
    '''
    judge if line (v1,v2) intersects with line(v3,v4)
    '''
    d = (v4[1]-v3[1])*(v2[0]-v1[0])-(v4[0]-v3[0])*(v2[1]-v1[1])
    u = (v4[0]-v3[0])*(v1[1]-v3[1])-(v4[1]-v3[1])*(v1[0]-v3[0])
    v = (v2[0]-v1[0])*(v1[1]-v3[1])-(v2[1]-v1[1])*(v1[0]-v3[0])
    if d<0:
        u,v,d= -u,-v,-d
    return (0<=u<=d) and (0<=v<=d)

def point_in_triangle2(A,B,C,P):
    v0 = [C[0]-A[0], C[1]-A[1]]
    v1 = [B[0]-A[0], B[1]-A[1]]
    v2 = [P[0]-A[0], P[1]-A[1]]
    cross = lambda u,v: u[0]*v[1]-u[1]*v[0]
    u = cross(v2,v0)
    v = cross(v1,v2)
    d = cross(v1,v0)
    if d<0:
        u,v,d = -u,-v,-d
    return u>=0 and v>=0 and (u+v) <= d

def tri_intersect2(t1, t2):
    '''
    judge if two triangles in a plane intersect 
    '''
    if line_intersect2(t1[0],t1[1],t2[0],t2[1]): return True
    if line_intersect2(t1[0],t1[1],t2[0],t2[2]): return True
    if line_intersect2(t1[0],t1[1],t2[1],t2[2]): return True
    if line_intersect2(t1[0],t1[2],t2[0],t2[1]): return True
    if line_intersect2(t1[0],t1[2],t2[0],t2[2]): return True
    if line_intersect2(t1[0],t1[2],t2[1],t2[2]): return True
    if line_intersect2(t1[1],t1[2],t2[0],t2[1]): return True
    if line_intersect2(t1[1],t1[2],t2[0],t2[2]): return True
    if line_intersect2(t1[1],t1[2],t2[1],t2[2]): return True
    inTri = True 
    inTri = inTri and point_in_triangle2(t1[0],t1[1],t1[2], t2[0])
    inTri = inTri and point_in_triangle2(t1[0],t1[1],t1[2], t2[1])
    inTri = inTri and point_in_triangle2(t1[0],t1[1],t1[2], t2[2])
    if inTri == True: return True
    inTri = True
    inTri = inTri and point_in_triangle2(t2[0],t2[1],t2[2], t1[0])
    inTri = inTri and point_in_triangle2(t2[0],t2[1],t2[2], t1[1])
    inTri = inTri and point_in_triangle2(t2[0],t2[1],t2[2], t1[2])
    if inTri == True: return True
    return False