Problem
GEO-B3-M03-P024 Dual Check of Pascal
B. New Original Problem. Let \(A_1,A_2,\ldots,A_6\) be six points on a circle \(\omega\), and let \(a_i\) be the tangent to \(\omega\) at \(A_i\). Define \(V_i=a_i\cap a_{i+1}\) modulo \(6\). Using polarity, prove that if Pascal gives the collinearity of the three intersections of opposite sides of the hexagon \(A_1A_2\ldots A_6\), then the lines \(V_1V_4\), \(V_2V_5\), \(V_3V_6\) are concurrent.
C. Hint 1. Identify which objects are poles of the Pascal points.
D. Hint 2. Collinearity of three points is dual to concurrence of their polars.
E. Full Solution.
Let \(P\) be the intersection of the sides \(A_1A_2\) and \(A_4A_5\). The polar of \(P\) passes through the poles of these two sides, namely through \(V_1=a_1\cap a_2\) and \(V_4=a_4\cap a_5\). Therefore the polar of \(P\) is the line \(V_1V_4\).
Similarly, the polars of the other two Pascal points are \(V_2V_5\) and \(V_3V_6\). By Pascal, the three Pascal points are collinear. The pole of this line lies on the polars of all three points. Hence the lines \(V_1V_4\), \(V_2V_5\), and \(V_3V_6\) pass through one point.
This is the closing problem of the module: it checks the full duality between Pascal and Brianchon.