Given a drawing of a parabola is there any geometric construction one can make to find its focus?

Question

This question was inspired by another one I asked myself these days Given a drawing of an ellipse is there any geometric construction we can do to find it's foci?

I think this is harder, I can't even find is axis or vertex.

Answer 1

Let three tangents to a parabola form a triangle. Then Lambert's theorem states that the focus of the parabola lies on the circumcircle of the triangle.

So draw three tangents. Find the triangle that they form and construct the circumcircle.

Start again with two of the original tangents and a new tangent. Find the triangle they form and construct the circumcircle. The focus must be at one of the two intersections of these circles.

Answer 2

As @AchilleHui mentions, the midpoints of two parallel chords lead to the point of tangency ($T$) with a third parallel line. Note that the line of midpoints is parallel to the axis of the parabola. By the reflection property of conics, the line of midpoints and the line $\overleftrightarrow{TF}$ make congruent angles with the tangent line.

So, two sets of parallel chords determine two points of tangency and two lines-of-midpoints, which determine two lines that meet at focus $F$. $\square$