Where I visualize cyclic ness?

[Fm Jakaria]

Let $ABC$ be an acute angled triangle with altitudes $AD, BE$ and $CF$. Suppose $EF$ intersects $BC$ at $P$. Let the line through $D$ parallel to $EF$ intersect $AB$ at $R$ and $AC$ at $Q$. Prove that the circumcircle of triangle $PQR$ passes through the midpoint of side $BC$.

[photon]

$\angle RQE = \angle QEP = \angle AEF = \angle ABC$
$\angle RQC = \angle RBC$ , $B,R,C,Q$ are cycic . $\therefore RD.DQ=BD.DC$
Let M be the midpoint of $BC$ . $M,D,E,F$ lie in nine-point circle . so , $PE.PF=PD.PM$
$\displaystyle \Rightarrow PC.PB = PD.PM$ [$PE.PF=PC.PB$ as $B,F,E,C$ are cyclic]
$\displaystyle \Rightarrow \frac{PM}{PC}=\frac{PB}{PD}$ $\displaystyle \Rightarrow \frac{MC}{PC}=\frac{BD}{PD}$
$\displaystyle \Rightarrow \frac{BM}{PC}=\frac{BD}{PD}............(1)$
$\displaystyle \Rightarrow \frac{BM}{BD}=\frac{PC}{PD} \Rightarrow \frac{BD}{BM}=\frac{PD}{PC} \Rightarrow \frac{MD}{BM}=\frac{CD}{PC} \Rightarrow \frac{MD}{CD}=\frac{BM}{PC}............(2)$
from (1),(2) - $\displaystyle \frac{MD}{CD}=\frac{BD}{PD}$
$MD.PD=BD.CD \Rightarrow MD.PD=RD.DQ$
so $M,Q,P,R$ are cyclic.

[*Mahi*]

Previously posted: viewtopic.php?f=13&t=2622 here.
You can check out further discussions there.

[zadid xcalibured]

[tanmoy]

why $\angle AEF=\angle ABC$?