## BAYMAX is cyclic!

For discussing Olympiad level Geometry Problems
Ananya Promi
Posts: 36
Joined: Sun Jan 10, 2016 4:07 pm
Location: Naogaon, Bangladesh

### BAYMAX is cyclic!

Let \$M\$ be the midpoint of the side \$AC\$ of triangle \$ABC\$. Let \$P\$ on \$AM\$ and \$Q\$ on \$CM\$ be such that \$PQ=AC/2\$. Let \$(ABQ)\$ intersect with \$BC\$ at \$A_X\$ other than \$B\$. \$(BCP)\$ intersects with \$BA\$ at \$A_Y\$ other than \$B\$. Prove that \$BA_YMA_X\$ is cyclic

Ananya Promi
Posts: 36
Joined: Sun Jan 10, 2016 4:07 pm
Location: Naogaon, Bangladesh

### Re: BAYMAX is cyclic!

Let \$N\$ and \$R\$ be the midpoints of \$AB\$ and \$BC\$ resp.
Then we get \$NRQP\$ is a parallelogram.
Again, \$AA_Y.AB=AP.AC\$ means, \$AA_Y.AN=AP.AQ\$
It gives \$NA_YPQ\$ cyclic.
Similarly, \$QMRA_X\$ is cyclic.
We are working with directed angle
\$\angle{APN}=-\angle{AA_YM}\$
Again, \$\angle{APN}=\angle{AQR}=\angle{MQR}=\angle{MA_XR}\$
So, \$\angle{MA_XR}=-\angle{AA_YM}\$
\$\angle{MA_XB}=\angle{MA_YB}\$
So, \$BA_YMA_X\$ is cyclic 