Let
![\triangle ABC](/media/m/1/f/3/1f3c3c0f3e134a169655f9511ba6ea82.png)
be an acute-angled triangle with
![AB \not= AC](/media/m/6/3/2/632558d83325fc1c84f72c2d70761e12.png)
. Let
![H](/media/m/4/c/0/4c0872a89da410a25f00b86366efece7.png)
be the orthocenter of triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
, and let
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
be the midpoint of the side
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
. Let
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
be a point on the side
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
and
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
a point on the side
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
such that
![AE=AD](/media/m/2/9/d/29da0e8b0834293d777d60bef18f39ae.png)
and the points
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
,
![H](/media/m/4/c/0/4c0872a89da410a25f00b86366efece7.png)
,
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
are on the same line. Prove that the line
![HM](/media/m/7/a/a/7aac075dd2e353efaf3631d84e2f84cb.png)
is perpendicular to the common chord of the circumscribed circles of triangle
![\triangle ABC](/media/m/1/f/3/1f3c3c0f3e134a169655f9511ba6ea82.png)
and triangle
![\triangle ADE](/media/m/a/9/7/a97712c9630d2cde9a78e98ab54296e8.png)
.
%V0
Let $\triangle ABC$ be an acute-angled triangle with $AB \not= AC$. Let $H$ be the orthocenter of triangle $ABC$, and let $M$ be the midpoint of the side $BC$. Let $D$ be a point on the side $AB$ and $E$ a point on the side $AC$ such that $AE=AD$ and the points $D$, $H$, $E$ are on the same line. Prove that the line $HM$ is perpendicular to the common chord of the circumscribed circles of triangle $\triangle ABC$ and triangle $\triangle ADE$.