You can not select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
This file contains ambiguous Unicode characters that may be confused with others in your current locale. If your use case is intentional and legitimate, you can safely ignore this warning. Use the Escape button to highlight these characters.
** 结论**：
$tan \alpha=\frac{1}{2},tan \beta=\frac{1}{3},\alpha+\beta=45^{\circ}$
利用结论：
设$\angle BAE=\angle \alpha,\angle FAD=\angle \beta$
$\because BE=2,AB=4$
$\large \therefore \frac{BE}{AB}=\frac{2}{4}=\frac{1}{2}=tan \alpha$
双$\because \alpha+ \beta=45^{\circ}$
根据结论：$\frac{DF}{AD}=\frac{1}{3} $
$\therefore DF=2$
** 证明**：
绘制右侧的图形，$\angle \alpha +\beta=45^{\circ}$
取角分线上任意一点$P$,向$AB,AC$引垂线，构造直角三角形，同时，延长$NP$交$AB$与$M$。
这样就有了好多个直角三角形，计算起来就容易了：
$\because tan\alpha=\frac{1}{2}$
设$EP=x$,则$AE=2x$
$\because \angle ANM$是引垂线引出的直角三角形，$\angle NAB=45^{\circ}$
$\therefore \angle AMN=45^{\circ}$
$\therefore EM=EP=x$
$PM=\sqrt{2}x,AM=3x$
再利用等腰直角三角形的边长关系，得到
$AN=MN$
$\because 2 AN^2=(3x)^2$
$AN=\sqrt{\frac{(3x)^2*2}{2*2}}=\frac{3\sqrt{2}}{2}x$
$\therefore PN=\frac{3\sqrt{2}}{2}x-\sqrt{2}x=\frac{\sqrt{2}}{2}x$
$\therefore tan \beta=\frac{PN}{AN}=\frac{1}{3}$
** 证毕**
