# Given both foci, a point on an ellipse and a random secant line how can one construct the meeting of the curves

We are given three points: $$F_1, F_2$$ that are the two foci of the ellipse $$\mathcal E$$ and point $$A$$ that belongs to $$\mathcal E$$.

We are also given a random line $$\ell$$ (let's assume it does not contain $$A$$).

How can we find points $$B$$ and $$C$$ such that $$\{B,C\} = \mathcal E \cap \ell \neq \emptyset$$ using straight edge and compass?

We can easily get:

• center of $$\mathcal E$$ as it is the midpoint of $$F_1F_2$$
• axes of symetry of $$\mathcal E$$
• major axis:

$$X = \odot(A,AF_1) \cap \overleftrightarrow{AF_2}$$ (with $$X$$ and $$F_2$$ in disjointed halfplanes of $$AF_1$$).

if $$M$$ is the midpoint of $$XF_2$$, then $$a = F_2M$$ so we have the vertices of the ellipse.

Construct centre and vertices of the ellipse, as explained in the question, as well as semi-axes $$a$$ and $$b$$. Let $$Q$$ be the point of intersection between lines $$\ell$$ and $$F_1F_2$$. Take any point $$P$$ on $$\ell$$ (different from $$Q$$) and its projection $$H$$ on $$F_1F_2$$. Construct then point $$P'$$ on ray $$HP$$ such that $$P'H:PH=a:b$$; line $$\ell'=QP'$$ is the image of $$\ell$$ under a dilation of ratio $$a/b$$ perpendicular to the major axis of the ellipse. (If $$\ell$$ is parallel to $$F_1F_2$$, then $$\ell'$$ is parallel to $$\ell$$ and can be constructed in the obvious way).
Construct then the auxiliary circle, centred at the centre of the ellipse and with radius $$a$$: this circle, too, is the image of the ellipse under a dilation of ratio $$a/b$$ perpendicular to the major axis of the ellipse. If $$B'$$ and $$C'$$ are the intersections of this circle with line $$\ell'$$, we can then easily construct $$B$$ and $$C$$ as the intersections of $$\ell$$ with the perpendiculars from $$B'$$ and $$C'$$ to the major axis of the ellipse.