# Euclidean geometry problems - mostly Menelaus' theorem

• Oct 19th 2010, 11:59 PM
jbpellerin
Euclidean geometry problems - mostly Menelaus' theorem
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