Name and/or generalization? “The slope of the secant through two points of a quadratic is the average of the slopes of the tangents at those points.”

Question

Does the following property of quadratic equations have a name? Is it generalized in some way? Or generalized to other functions?

Pick any two points on the graph of any quadratic. Draw a secant line through the points. Draw a tangent line through each of the points. The slope of the secant line will equal the arithmetic average of the slopes of two tangent lines.

I have examined the wikipedia pages for quadratics and parabolas and there seems to be a lot of similar properties but (perhaps) not exactly this one. That is why I think it must be a specific case of something more general or maybe it is so obvious that it is not mentioned. It seems to be related to the mean value theorem as well.

Answer 1

This is related to a well-known property of parabolas:

The line parallel to the axis through the midpoint of a chord $AB$, also passes through the intersection of the tangents at $A$ and $B$.

As you can see in figure below, we have then:

$$ m_{AC}=-{y_A\over x},\quad m_{BC}={y_B\over x},\quad m_{AB}={y_B-y_A\over 2x}. $$