Problem
GEO-B3-M03-P016 Contact Chords from Three Vertices
B. New Original Problem. Triangle \(ABC\) is circumscribed about a circle \(\omega\). For each vertex, join the two contact points of the sides issuing from that vertex. This gives three lines \(a,b,c\). Prove that the poles of \(a,b,c\) are respectively the vertices \(A,B,C\).
C. Hint 1. At each vertex, the two sides are tangents to \(\omega\).
D. Hint 2. The line through the two contact points of these tangents is the polar of the vertex.
E. Full Solution.
Consider vertex \(A\). The sides \(AB\) and \(AC\) touch \(\omega\) at two points. The line joining these contact points is, by definition, the polar of \(A\). Hence its pole is \(A\).
The same argument applies to vertices \(B\) and \(C\). Therefore the poles of the three constructed lines are \(A,B,C\), respectively.
The task looks simple, but it is needed for later dual proofs in tangential polygons.