![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
is an acute-angled triangle.
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
is the midpoint of
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
and
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
is the point on
![AM](/media/m/9/2/1/921d54bb92ada2d2120b2591b722ea12.png)
such that
![MB = MP](/media/m/e/8/5/e858bb47831fe9f3ad98849f474c4f3a.png)
.
![H](/media/m/4/c/0/4c0872a89da410a25f00b86366efece7.png)
is the foot of the perpendicular from
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
to
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
. The lines through
![H](/media/m/4/c/0/4c0872a89da410a25f00b86366efece7.png)
perpendicular to
![PB](/media/m/7/1/c/71c3da8e488f46729e3d8d507af4af81.png)
,
![PC](/media/m/8/a/5/8a52c00b6f337972533724ba744a62da.png)
meet
![AB, AC](/media/m/d/7/6/d769eff805676f4c17bb0624e6a4ccef.png)
respectively at
![Q, R](/media/m/5/0/8/508477b1a6cfe7e08c0fc60efd4088a5.png)
. Show that
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
is tangent to the circle through
![Q, H, R](/media/m/1/e/4/1e45e26d0289f818f86940c777a62009.png)
at
![H](/media/m/4/c/0/4c0872a89da410a25f00b86366efece7.png)
.
Original Formulation:
For an acute triangle
![ABC, M](/media/m/0/8/6/0862c6d5c55b6b86f8bf27172445fa35.png)
is the midpoint of the segment
![BC, P](/media/m/f/f/e/ffefda56a2fb93913f5264acd6f1ede3.png)
is a point on the segment
![AM](/media/m/9/2/1/921d54bb92ada2d2120b2591b722ea12.png)
such that
![PM = BM, H](/media/m/9/5/b/95bc4fb735c45b34fb7815bf94408e10.png)
is the foot of the perpendicular line from
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
to
![BC, Q](/media/m/c/2/b/c2b650b32d9c422d7a94b123b7ba7f28.png)
is the point of intersection of segment
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
and the line passing through
![H](/media/m/4/c/0/4c0872a89da410a25f00b86366efece7.png)
that is perpendicular to
![PB,](/media/m/d/a/3/da3018460e07c17465e8b3df37d21a42.png)
and finally,
![R](/media/m/4/d/7/4d76ce566584cfe8ff88e5f3e8b8e823.png)
is the point of intersection of the segment
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
and the line passing through
![H](/media/m/4/c/0/4c0872a89da410a25f00b86366efece7.png)
that is perpendicular to
![PC.](/media/m/b/3/4/b341df0332c7815338950e8808ce9537.png)
Show that the circumcircle of
![QHR](/media/m/4/0/d/40dc57643d09bdac1951ca4762644ccd.png)
is tangent to the side
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
at point
%V0
$ABC$ is an acute-angled triangle. $M$ is the midpoint of $BC$ and $P$ is the point on $AM$ such that $MB = MP$. $H$ is the foot of the perpendicular from $P$ to $BC$. The lines through $H$ perpendicular to $PB$, $PC$ meet $AB, AC$ respectively at $Q, R$. Show that $BC$ is tangent to the circle through $Q, H, R$ at $H$.
Original Formulation:
For an acute triangle $ABC, M$ is the midpoint of the segment $BC, P$ is a point on the segment $AM$ such that $PM = BM, H$ is the foot of the perpendicular line from $P$ to $BC, Q$ is the point of intersection of segment $AB$ and the line passing through $H$ that is perpendicular to $PB,$ and finally, $R$ is the point of intersection of the segment $AC$ and the line passing through $H$ that is perpendicular to $PC.$ Show that the circumcircle of $QHR$ is tangent to the side $BC$ at point $H.$