AoPS Academy
are 
and 
Find the product of all possible values of 
![\[ p = \frac{2ab}{a + b} \]](http://latex.artofproblemsolving.com/0/f/4/0f4297ec59fe97c92fda75af4cc835be6b9a2507.png)
, ![\[ p, a, b \in \mathbb{C} \]](http://latex.artofproblemsolving.com/b/4/4/b4499970ac49d770a810d7345fedef7773cd03f5.png)
such that there exists a pair 
of positive integers, such that 
is not divisible by the cube of any prime, and 
are constructed on the exterior of a triangle 
. The perpendiculars from the midpoints of 
to the respective lines 
are constructed. Prove that these three perpendiculars pass through a single point.
are constructed on the exterior of a triangle . The perpendiculars from the midpoints of to the respective lines are constructed. Prove that these three perpendiculars pass through a single point.
be the reflection of 
across line 
Since 
we deduce that 
and because of 
and 
it follows that 
by SAS criterion 
Similar reasoning yields 
is a parallelogram, thus 
and 
bisect each other at 
If 
denote the foot of the A-altitude and midpoint of 
then perpendicular 
from 
to 
is the midline of the trapezoid 
passes through the 9-point center of 
Likewise, perpendiculars from midpoints of 
to 
pass through the 9-point center of 
denotes the sides of triangle as usual. Let 
be the midpoints of 
, respectively. In 
, 
is a median 
. Similarly we have 
.
, and we are done by Carnot's theorem.
