The altitudes through the vertices
![A,B,C](/media/m/6/0/1/6012c28979f41c54e9b40b9fc855aa34.png)
of an acute-angled triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
meet the opposite sides at
![D,E, F,](/media/m/7/3/9/73990275e679f52e8275bc761abf0fb5.png)
respectively. The line through
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
parallel to
![EF](/media/m/f/5/5/f5594d5ec47ea777267cf010e788fedd.png)
meets the lines
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
and
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
at
![Q](/media/m/4/5/c/45ce8d14aa1eb54f755fd8e332280abd.png)
and
![R,](/media/m/8/0/c/80cf2824b1a982d0be339aca38802438.png)
respectively. The line
![EF](/media/m/f/5/5/f5594d5ec47ea777267cf010e788fedd.png)
meets
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
at
![P.](/media/m/9/5/c/95c3cecafab8ed471d070cfcc15c8428.png)
Prove that the circumcircle of the triangle
![PQR](/media/m/e/d/5/ed5efbddbf500aeff476d02507a9f80a.png)
passes through the midpoint of
%V0
The altitudes through the vertices $A,B,C$ of an acute-angled triangle $ABC$ meet the opposite sides at $D,E, F,$ respectively. The line through $D$ parallel to $EF$ meets the lines $AC$ and $AB$ at $Q$ and $R,$ respectively. The line $EF$ meets $BC$ at $P.$ Prove that the circumcircle of the triangle $PQR$ passes through the midpoint of $BC.$