Problem
GEO-B3-M04-P023 Envelope of Simson Lines
B. New Original Problem. A point \(P\) moves on the circumcircle of triangle \(ABC\). Prove that the family of its Simson lines has an envelope: each Simson line is tangent to a fixed curve. Indicate how the contact point is constructed through the midpoint of \(PH\).
C. Hint 1. Use the fact that the Simson line passes through the midpoint of \(PH\).
D. Hint 2. As \(P\) moves, this midpoint moves on the nine-point circle, while the line direction rotates at half speed.
E. Full Solution.
Let \(M\) be the midpoint of \(PH\). Then \(M\) lies on the nine-point circle and also on the Simson line of \(P\). Under a small motion of \(P\), the point \(M\) moves on the nine-point circle, while the direction of the Simson line changes in a way coordinated with the tangent direction of the path of \(M\).
The contact point of the envelope is obtained as the limiting position of the intersection of two nearby Simson lines. It can be constructed from \(M\): draw the Simson line through \(M\) and use that the angular speed of this line is half the angular speed of \(P\). The resulting envelope is the classical Steiner deltoid.
For this module, the important part is not the name of the curve, but the mechanism: the Simson line always passes through a point of the nine-point circle, while its direction changes so that a fixed envelope appears.
This is a survey-style level 5 task, preparing for later topics on the Steiner deltoid.