Putnam 2001, Problem A-4

Triangle $ABC$ has an area $1$. Points $E, F, G$ lie, respectively, on sides $BC, CA, AB$ such that $AE$ bisects $BF$ at point $R$, $BF$ bisects $CG$ at point $S$, and $CG$ bisects $AE$ at point $T$. Find the area of the triangle $RST$.

Solution:
Choose $r,s,t$ so that $EC = rBC, FA = sCA, GB = tCB$, and let $[XYZ]$ denote the area of triangle $XYZ$. Then $[ABE] = [AFE]$ since the triangles have the same altitude and base. Also $[ABE] = (BE/BC)[ABC] = 1-r$, and $[ECF] = (EC/BC)(CF/CA)[ABC] = r(1-s)$ (e.g., by the law of sines). Adding this all up yields
$$1 = [ABE] + [ABF] + [ECF]$$
$$= 2(1-r) + r(1-s) = 2-r-rs$$

or $r(1+s) = 1$. Similarly $s(1+t) = t(1+r) = 1$.

Let $f: [0,\infty) \to [0,\infty)$ be the function given by $f(x) = 1/(1+x)$; then $f(f(f(r))) = r$. However, $f(x)$ is strictly decreasing in $x$, so $f(f(x))$ is increasing and $f(f(f(x)))$ is decreasing. Thus there is at most one $x$ such that $f(f(f(x))) = x$; in fact, since the equation $f(z) = z$ has a positive root $z = (-1 + \sqrt{5})/2$, we must have $r = s = t = z$.

We now compute $[ABF] = (AF/AC)[ABC] = z$, $[ABR] = (BR/BF)[ABF] = z/2$, analogously $[BCS] = [CAT] = z/2$, and $[RST] = |[ABC] - [ABR] - [BCS] - [CAT]| = |1-3z/2| = \frac{7-3\sqrt{5}}{4}$.

Note: the key relation $r(1+s) = 1$ can also be derived by computing using homogeneous coordinates or vectors.