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.

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  • $\begingroup$ You mean $JX$ goes through $MN$ at $G$ and $G$ is midpoint of $MN$? $\endgroup$ – Word Shallow Jan 1 at 4:27
  • $\begingroup$ Yes, of course. help me show this pls. $\endgroup$ – Quỳnh Vũ Thị Jan 1 at 5:42

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 $(Q.E.D)$

  • $\begingroup$ What about exercise b? Thank you. $\endgroup$ – Quỳnh Vũ Thị Jan 2 at 3:50

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