3. The incircle of $\displaystyle \Delta ABC$ touches the sides $\displaystyle BC, CA, AB$ at the points $\displaystyle X,Y,Z$ respectively, and $\displaystyle YZ$ is produced to meet $\displaystyle BC$ at $\displaystyle K$

Show that

$\displaystyle {BX}/{CX} = {BK}/{CK}$

4.A circle meets side $\displaystyle BC$ of $\displaystyle \Delta ABC$ at points $\displaystyle L$ and $\displaystyle L'$, $\displaystyle CA$ at $\displaystyle M$ and $\displaystyle M'$, and $\displaystyle AB$ at $\displaystyle N$ and $\displaystyle N'$. Show that if $\displaystyle AL, BM, CN$ are concurrent, then $\displaystyle AL', BM', CN'$ are also concurrent.

6.In $\displaystyle \Delta ABC$, let $\displaystyle E$ and $\displaystyle F$ be points on the sides $\displaystyle AC$ and $\displaystyle AB$, respectively, with $\displaystyle AE = AF$. If

the median $\displaystyle AD$ intersects the segment $\displaystyle EF$ at the point $\displaystyle P$, show that

$\displaystyle {PE}/{AB} = {PF}/{AC}$

I'm pretty sure they all just require creative use of Menelaus' theorem, but I can't seem to figure out in which ways

I don't want the questions solved entirely for me, maybe just some hints (or rot13 the answer so I am encouraged to do it myself before checking)

Turns out I had to use power of a point as well, done now