Locus of a point on the side of a square
by EmersonSoriano, Apr 2, 2025, 9:58 PM
Let 
be a fixed square and 
a variable point on segment 
. The square 
is constructed such that 
is on segment 
and 
is on segment 
. Let 
be the intersection point of lines 
and 
. Find the locus of as varies along segment .
be a fixed square and 
a variable point on segment 
. The square 
is constructed such that 
is on segment 
and 
is on segment 
. Let 
be the intersection point of lines 
and 
. Find the locus of as varies along segment .
be an acute-angled triangle, with circumcenter 
,
and orthocenter 
.
be a point on 
such that 
.
and 
similarly.
,
be an acute triangle with 
,
be the projection from 
onto 
be a point on the extension of 
.
be on the perpendicular bisector of 
such that ![\[\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ\]](http://latex.artofproblemsolving.com/c/6/3/c63eeb053f1fa8a29bef93516814ae121382219e.png)
Let the reflection of 
and 
be 
and 
,
and 
such that 
.
and 
intersect the circumcircles of 
and 
at 
and 
and 
intersect 
and 
at 
and 
.
intersect 
.
.
and that line 
is not parallel to line 
is perpendicular to 
.
is a diameter, and 
and 
are parallel. Let 
be the midpoint of 
.
and 
are perpendicular.
and 
are perpendicular.
be the circumcircle of triangle 
at 
respectively. 
and 
is the intersection of 
,
,
and that 
,
.
.
externally bisect 
and let 
be the line passing through 
at point 
meet line 
at point 
and the internal bisector of 
at point 
.
are concyclic.
.
and 
,
is not less than the area of 
be an acute triangle. Denote by 
the foot of the perpendicular line drawn from the point 
to the side 
,
the orthocenter of 
be the point of intersection of the circumcircle 
of the triangle 
,
be the point of intersection (other than 
and the circle 
must hold.
the length of the line segment 
April 2021
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