2, 4), then the vertex of the parabola is (2, 3).

To see why this is true, let's use the fact that the tangent to a parabola at a point P is the line that passes through P and is parallel to the parabola's axis of symmetry.

First, let's find the axis of symmetry of the parabola that passes through (1, 2) and (3, 6). The midpoint of the segment connecting these two points is (2, 4), which we know lies on the axis of symmetry.

Next, let's find the slope of the tangent at (1, 2). Since the tangent is parallel to the axis of symmetry, it must have the same slope as the line passing through (2, 4) and (1, 2). This line has slope (4-2)/(2-1) = 2. Therefore, the tangent at (1, 2) has slope -1/2 (since the tangent is perpendicular to the line connecting (1, 2) and (2, 4)).

Similarly, the tangent at (3, 6) has slope -1/2 (since it is parallel to the tangent at (1, 2)).

Now we can use the fact that the vertex of a parabola lies at the midpoint of the segment connecting its two points of intersection with any tangent line. Let's find the points where the two given tangents intersect the y-axis (i.e. where x=0).

For the tangent at (1, 2), we have y - 2 = (-1/2)(x - 1), so y = (-1/2)x + 5/2. Therefore, the point of intersection with the y-axis is (0, 5/2).

For the tangent at (3, 6), we have y - 6 = (-1/2)(x - 3), so y = (-1/2)x + 9/2. Therefore, the point of intersection with the y-axis is (0, 9/2).

The vertex of the parabola must lie at the midpoint of the segment connecting (0, 5/2) and (0, 9/2), which is (0, 7/2). But we know that the vertex also lies on the axis of symmetry, which passes through (2, 4). Therefore, the vertex must be (2, 3), which is the midpoint of the segment connecting (0, 7/2) and (4, 0).