Tangent slope

I just had a four-hour-straight math test, and I found one of the questions rather interesting, at least interesting enough to share with you. I found a pretty great solution to it, and I'm interested to see what you guys can come up with.

Determine (and prove) whether or not a secant of a second-degree function on the form (y=ax^2+bx), which always goes through origo and one more point on the curve, has the same slope as a tangent of the curve which tangents the curve on the arithmetic mean value of the two x values of the secant. The curve always has a vertex (extreme point) in origo.