omg i haven't done geo in a month
by wu2481632, Nov 4, 2016, 3:08 PM
so rusty
Let
and
be the circumcircle and the circumcentre of an acute-angled triangle
with
. The angle bisector of
intersects
at
. Let
be the circle with diameter
. The angle bisectors of
and
intersect
at points
and
respectively. The point
is chosen on the line
so that
. Prove that
. (ISL 2014 G3)
Solution
Let


















Solution
darn i had to take a hint for the last paragraph
Firstly, the angle bisectors of
and
are simply the perpendicular bisectors of
and
.
Denote by
and
the second intersections of the perpendicular bisectors of
and
with
.
Lemma:
is an isosceles trapezoid.
Proof: It suffices to prove that the angles between line
and lines
and
are equivalent (by symmetry), as
lies on the perpendicular bisector of
. But this is obvious because
is perpendicular to the angle bisector of
.
Now, as
is the intersection of the diagonals of this quadrilateral, we know that
. But this implies
.
Denote by
the midpoint of
. It is clear that
lies on
.
Lemma:
is cyclic.
Proof: Firstly,
is the center of
. Next, we see that
. Because
is the aforementioned center, we get
. Summing those arcs yields
. However, now we just need to add
to this sum, which does indeed give us the full
, so
is indeed cyclic.
Now consider a point
such that
and
lies on
. As
and
, it follows that
. From this we deduce that
is cyclic, and that
is the radical axis of
and
. Furthermore,
is the radical axis of
and
, so the radical center of those three circles is simply
, implying that
lies on
, so we're done.
Firstly, the angle bisectors of




Denote by





Lemma:

Proof: It suffices to prove that the angles between line







Now, as



Denote by




Lemma:

Proof: Firstly,









Now consider a point
















