, 2014

Problem:
Pick a subset of at least four of the following geometric theorems, order them from earliest to latest by publication date, and write down their labels (a single capital letter) in that order. If a theorem was discovered multiple times, use the publication date corresponding to the geometer for which the theorem is named.

C. (Ceva) Three cevians $AD$, $BE$, $CF$ of a triangle $ABC$ are concurrent if and only if $\frac{BD}{DC} \frac{CE}{EA} \frac{AF}{FB}=1$.

E. (Euler) In a triangle $ABC$ with incenter $I$ and circumcenter $O$, we have $IO^{2}=R(R-2r)$, where $r$ is the inradius and $R$ is the circumradius of $ABC$.

H. (Heron) The area of a triangle $ABC$ is $\sqrt{s(s-a)(s-b)(s-c)}$, where $s=\frac{1}{2}(a+b+c)$.

M. (Menelaus) If $D, E, F$ lie on lines $BC, CA, AB$, then they are collinear if and only if $\frac{BD}{DC} \frac{CE}{EA} \frac{AF}{FB}=-1$, where the ratios are directed.

P. (Pascal) Intersections of opposite sides of cyclic hexagons are collinear.

S. (Stewart) Let $ABC$ be a triangle and $D$ a point on $BC$. Set $m=BD, n=CD, d=AD$. Then $man+dad=bmb+cnc$.

V. (Varignon) The midpoints of the sides of any quadrilateral are the vertices of a parallelogram.

If your answer is a list of $4 \leq N \leq 7$ labels in a correct order, your score will be $(N-2)(N-3)$. Otherwise, your score will be zero.