2025 USAMO Problems/Problem 4
- The following problem is from both the 2025 USAMO #4 and 2025 USAJMO #5, so both problems redirect to this page.
Contents
[hide]Problem
Let be the orthocenter of acute triangle
, let
be the foot of the altitude from
to
, and let
be the reflection of
across
. Suppose that the circumcircle of triangle
intersects line
at two distinct points
and
. Prove that
is the midpoint of
.
Solution 1
Let AP intersects BC at D. Extend FC to the point E on the circumcircle of
. Since
is the orthocenter of
, we know that
or
, and
. Next we use the power of H in
:
. These relations imply that
.
Hence are midpoints of
respectively. By midline theorem,
. Since
, we have
. This implies that
. Consequently,
is the diameter of
. Let
be the midpoint of
which is also the center of
.
are midpoints of
respectively. By the midline theorem again, we have
, consequently,
. This implies that
is the perpendicular bisector of the chord
hence
is the midpoint of
. ~ Dr. Shi davincimath.com
Solution 2
Denote as the center of
,
as the center of
,
as the midpoint of
,
as the midpoint of
, and
as the midpoint of
. It suffices to show that
.
Claim: is cyclic.
Proof: Since and
, KM is a midline of
and
.
as well since
, so
lies on
.
Next, note that
lies on
, so the perpendicular bisector of
through
passes through
. In other words,
, and
are collinear. Since
and
are both perpendicular to
, it follows that they are parallel.
Since
and
, then
.
Finally, we have that
and thus
is cyclic. It follows that
, so
, as desired.
-mop
Solution 3
Connect and have
intersect
at
. Also extend
past point
and have it intersect with the circle at point
.
Since is the reflection of
over
, we know that
. Since
is the orthocenter, we can draw the altitude and tell that
,
, and
are collinear.
We know
and
, so
by AA, so
.
and
. From this, we can tell that
. Therefore,
and
.
If we connect , we can tell that that
due to
being the reflection of
and
being perpendicular to
, so
. In addition,
. Also,
because they are vertical angles.
So, because of SAA. From this we can conclude that
, so
is the midpoint of
.
Solution 4
Let be the foot of the altitude from
to
By Power of a Point, we have
and
Adding, we get
It is well known that the reflection of
over
which we denote by
lies on
Then, let
We have
Thus,
and since
we have
Hence,
is the midpoint of
~TThB0501
Solution 5
Let Q be the antipode of B. Claim — AHQC is a parallelogram, and AP CQ is an isosceles trapezoid.
Proof. As AH ⊥ BC ⊥ CQ and CF ⊥ AB ⊥ AQ. Let M be the midpoint of QC.
Claim — Point M is the circumcenter of triangle AFP.
Proof. It’s clear that MA = MP from the isosceles trapezoid.
As for MA = MF, let N denote the midpoint of AF; then MN is a midline of the parallelogram, so MN ⊥ AF. Since CM ⊥ BC and M is the center of (AF P), it follows CX = CY .
Video Solution - A 2-minute proof
Solution 6
Quick angle chasing gives .
Let
be the circumcenter of
.
Thus (because O and A lie on the same side of segment
).
As , the quadrilateral FOCP is cyclic.
Observe that , so
.
From the properties of cyclic quadrilaterals,
Thus .
Therefore
is perpendicular to chord
,
.