Isotomic Lines

Given a point $P$ in the interior of a Triangle $\Delta A_1A_2A_3$, draw the Cevians through $P$ from each Vertex which meet the opposite sides at $P_1$, $P_2$, and $P_3$. Now, mark off point $Q_1$ along side $A_2A_3$ such that $A_3P_1=A_2Q_1$, etc., i.e., so that $Q_i$ and $P_i$ are equidistance from the Midpoint of $A_jA_k$. The lines $A_1Q_1$, $A_2Q_2$, and $A_3Q_3$ then coincide in a point $Q$ known as the Isotomic Conjugate Point.

See also Cevian, Isotomic Conjugate Point, Midpoint