Consider
![\triangle OAB](/media/m/b/0/6/b06bad4d5e1c25dd24d1602e4f78a89c.png)
with acute angle
![AOB](/media/m/3/1/c/31c3942c20efa169bde766e004ffeaa2.png)
. Thorugh a point
![M \neq O](/media/m/1/4/1/1419fd3af3621ae90be881801535167b.png)
perpendiculars are drawn to
![OA](/media/m/b/2/0/b206c115fb0e114a37cf644cba5338cb.png)
and
![OB](/media/m/5/0/3/503e9123196089d1244989e870075ca4.png)
, the feet of which are
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
and
![Q](/media/m/4/5/c/45ce8d14aa1eb54f755fd8e332280abd.png)
respectively. The point of intersection of the altitudes of
![\triangle OPQ](/media/m/3/a/e/3aeb56cec05bd4fb05ced957817277b8.png)
is
![H](/media/m/4/c/0/4c0872a89da410a25f00b86366efece7.png)
. What is the locus of
![H](/media/m/4/c/0/4c0872a89da410a25f00b86366efece7.png)
if
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
is permitted to range over
a) the side
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
;
b) the interior of
![\triangle OAB](/media/m/b/0/6/b06bad4d5e1c25dd24d1602e4f78a89c.png)
.
%V0
Consider $\triangle OAB$ with acute angle $AOB$. Thorugh a point $M \neq O$ perpendiculars are drawn to $OA$ and $OB$, the feet of which are $P$ and $Q$ respectively. The point of intersection of the altitudes of $\triangle OPQ$ is $H$. What is the locus of $H$ if $M$ is permitted to range over
a) the side $AB$;
b) the interior of $\triangle OAB$.