2017 AMC 12B 第 24 题

Set $BC = 1,$ $AB = r,$ and place $C = (0,0),$ $B = (0,1),$ $A = (r,1),$ $D = (\tfrac1r, 0)$

Find $E$ from $\triangle ABC \sim \triangle CEB,$ compute both areas, and set $[\triangle AED] = 17[\triangle CEB]$

Set $BC = 1$ and $AB = r \gt 1.$ The similarity $\triangle ABC \sim \triangle BCD$ with the right angles places the figure at $C = (0,0),$ $B = (0,1),$ $A = (r,1),$ $D = \bigl(\tfrac1r, 0\bigr).$ Let $E = (x, y)$ with $x, y \gt 0.$ From $\triangle ABC \sim \triangle CEB$ we get $\dfrac{x}{y} = \tan(\angle ECB)$ $= \tan(\angle BAC)$ $= \dfrac1r$ and $x^2 + y^2 = \dfrac{r^2}{1 + r^2},$ so $x = \dfrac{r}{1+r^2},$ $y = \dfrac{r^2}{1+r^2}.$ The two relevant areas are $$\begin{aligned} [\triangle CEB]&=\dfrac{r}{2(1+r^2)},\\ [\triangle AED]&=\dfrac{r^4-r^2+1}{2r(1+r^2)}. \end{aligned}$$ Setting the second equal to $17$ times the first gives $r^4 - 18r^2 + 1 = 0.$ Then $r^2 = 9 + 4\sqrt5 = (2 + \sqrt5)^2,$ so $r = 2 + \sqrt5.$

Thus, the correct answer is D.