Given triangle ABC, parallel lines DC and EA are reflected in BC and BA respectively.
What is the locus of F, the intersection of the reflected lines?
(This problem comes from Alex Turzillo's project entitled “Geometric Measure of Aberration in Parabolic Caustics”).

Where do you put the control points of a cubic spline in order for the spline to be a piece of a parabola?
Explore here. Prove using Geometry Expressions (or by hand).

The circle isotomic with respect to P is the locus of the reflection of P in the tangents of the circle.
The Conchoid is the locus of the points a given fixed distance from a point on the circle, and lying on a line through that point and P.

A is the circle center, and |AB| = |BD|.
What is the relationship between angle CAE and angle BDA?
You can drag B and C and watch it change.
Can you prove the relationship?
This is an example from Archimedes Book of Lemmas.