Triangle $ABC$ has an incircle $(I)$ which contacts $BC,CA,AB$ at $D,E,F$. On line $EF$ we get two points $M$ and $N$ such that $CM//BN//AD$. $DM$ and $DN$ cut $(I)$ at $P,Q$.
a, Prove that $BP,CQ,AD$ concur.
b, Let $J$ be point which $BP,CQ,AD$ concur. $X$ is midpoint of $PQ$. Show that $JX$ intersects $MN$ at the midpoint $G$ of $MN$.
I don't know which lemmas we use(maybe Ceva theorem, Thales theorem because there are three paralel lines). Show please and anyone can tell me some geometry book for studying? Thank.
Best Answer
a) Let $S$ be the intersection of $EF$ and $BC$,the segment $AD$ and incircle $(I)$ be $T$; $J$ be the intersection of $SP$ and $AD$; $BQ$ intersects $CP$ at $V$ .Then we have $ST$ is tangent of incircle $(I)$
So the polar of $S$ is the line $AD$.
We have: $V(SJ,QP)=-1$ and $V(SD,BC)=-1$
And $VS\equiv VS,VB\equiv VQ,VC\equiv VP\Rightarrow VD\equiv VJ$
So $V\in AD$. In $\Delta VBC$: $P;Q$ are respectively in the $CV$ and $BV$
$PQ$ intersects $BC$ at $S$ and $(SD,BC)=-1$ so we have $MV;BQ;CQ$ concur.
Or $BP;CQ;AD$ concur $(Q.E.D)$