# Euclidean geometry problems - mostly Menelaus' theorem

• Oct 20th 2010, 12:59 AM
jbpellerin
Euclidean geometry problems - mostly Menelaus' theorem
3. The incircle of $\Delta ABC$ touches the sides $BC, CA, AB$ at the points $X,Y,Z$ respectively, and $YZ$ is produced to meet $BC$ at $K$
Show that

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

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

6.In $\Delta ABC$, let $E$ and $F$ be points on the sides $AC$ and $AB$, respectively, with $AE = AF$. If
the median $AD$ intersects the segment $EF$ at the point $P$, show that
${PE}/{AB} = {PF}/{AC}$

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