**Example 1:** Let $UV$ be a chord of an ellipse centered at $O$ such that ${OU}\perp{OV}$. Determine the locus of the foot $P$ of the perpendicular from $O$ to the chord $UV$ when $U$ moves along the ellipse.

The appropriate locus is a circle.

Experimenting with the position of point O it turns out that it may not be at the center of a conic.

This enables us to apply this construction to all conics, e.g. to a parabolla, as given below.

It also works for singular conics.

If the lines are mutually orthogonal, we get a line instead of a circle.

*Proof:* The case of two orthogonal lines can be proved straightforwardly. Applying the *Simson-Wallace theorem*
on the triangle *QUV* and *O* on its circumcircle, the points *P*, *S*, *R* are collinear
(the *Simson line*).

**Example 2:** (*Frégier point*) Given a conic *κ* and a point *O* on *κ*, then the hypotenuses of
right-angled triangles inscribed to *κ* and having common right-angle vertex *O* intersect at one
point *F* – the Frégier point to *O* with respect to *κ*.

*Proof:*
When *U* moves along *κ*, the point *P* forms a circle *c*.
Since the angle *OPU* is right, then also the angle *OPF* is right.
Then according to the theorem of Thales the point *F* must be fixed. It is the opposite point to the point *O* on the circle *c*.

*Relevant links:*

Mathoverflow: Frégier and Frégier's Theorem

Weiss, G. Frégier points revisited. *South Bohemia Mathematical Letters, Vol. 26 (2018), No. 1., 84-92.*

**Example 3:** Given a line *k*, a point *O* on it, let the line *l*
be perpendicular to *k* at *O* with a point *M* on it, and let *A* be an arbitrary point
in the plane. Determine the locus of the intersection *P* of the circle *c* centered at *M*
with radius *MO* with the line *AB*, where *B ∈ l* is the opposite point to *O*
in the circle *c*, when *M* moves along the line *l*.

If *M* moves along the line *l* then the locus of *P* is a circle. Try it using the dynamic figure below.

*Proof:* According to the *Thales' theorem*, the angle *BPO* is right. This implies that the angle *APO* is right as well.
Using the Thales' theorem once again, we come to the conclusion that the point *P* lies on a circle with a fixed diameter *|OA|*.