From the orthocenter to a projective closure theorem (Q1904119)

Let \(\Delta = p_1p_2p_3\) be a triangle, and assume \(p_i p_j \cap p_k s = q_k\). Then \(s\) is the orthocenter of \(\Delta\) if and only if the points \(q_i\), \(q_j\), \(p_k\), \(s\) are concircular for \(\{i, j, k\} = \{1,2,3\}\). The author considers the centers \(a_k\) of these circles, the midpoints \(b_k\) of \(p_k s\), the midpoints \(c_k\) of \(q_k s\), and the intersections \(d_k = a_i a_j \cap a_k b_k\) on the line at infinity. This leads him to the following configurational theorem of rank 10 on 12 points and 10 lines holding in any Desarguesian projective plane: If \(a_k b_k d_k\), \(a_i a_j c_k d_k\), \(b_i b_j c_k\), and \(d_1 d_2 d_3\) are collinear for \(\{i,j,k\} = \{1,2,3\}\), then the 3 lines \(b_k c_k\) intersect in a common point \(s\).