Problem
GEO-B3-M03-P003 Checking La Hire's Theorem
#3
★☆☆☆☆ Level 1 of 5
B. New Original Problem. With respect to the unit circle, points \(P(p,q)\) and \(Q(u,v)\) are such that \(Q\) lies on the polar of \(P\). Prove that \(P\) lies on the polar of \(Q\).
Inspired by Prasolov poles and polars method
C. Hint 1. Write the condition \(Q\in p\) in coordinates.
D. Hint 2. Compare the equation of the polar of \(Q\) with the same scalar equality.
E. Full Solution.
The polar of \(P\) has equation \(px+qy=1\). Since \(Q(u,v)\) lies on it, we get \(pu+qv=1\).
The polar of \(Q\) has equation \(ux+vy=1\). Substituting \(P(p,q)\), we get \(up+vq=1\), the same equality. Therefore \(P\) lies on the polar of \(Q\).
After this task, La Hire may be used as a ready tool without coordinates.