is an acute-angled triangle.
is the midpoint of
and
is the point on
such that
.
is the foot of the perpendicular from
to
. The lines through
perpendicular to
,
meet
respectively at
. Show that
is tangent to the circle through
at
.
Original Formulation:
For an acute triangle
is the midpoint of the segment
is a point on the segment
such that
is the foot of the perpendicular line from
to
is the point of intersection of segment
and the line passing through
that is perpendicular to
and finally,
is the point of intersection of the segment
and the line passing through
that is perpendicular to
Show that the circumcircle of
is tangent to the side
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 $H.$
Školjka