AoPS Academy
and 
, Jaqueline has a ruler that allows tracing the line 
. Jaqueline also has a special object that allows the construction of a circle of diameter . Also, always when two circles (or a circle and a line, or two lines) intersect, she can mark the points of the intersection with a pencil and trace more lines and circles using these dispositives by the points marked. Initially, she has an acute scalene triangle 
. Show that Jaqueline can construct the incenter of .
be triangle with point 
and 
on 
and 
are on the same side of 
be midpoint of segment and 
be midpoint of segment 
be intersection of 
with 
and 
that verify the following equation for all 
:
be positive real integer such that 
Prove that
such that 
and 
that satisfy the equation 
.
be a circle, and let a line 
intersect at two points, and 
Circles 
and 
are internally tangent to at points 
and 
respectively, and both are tangent to at a common point 
Similarly, circles 
and 
are externally tangent to at and respectively, and are tangent to at points 
and 
respectively. is 
the segment 
has a length of 
and 
find the length of segment 
,
,
, 
, 
is not greater than 
.
be the feet of the heights of an acute triangle 
On the segments 
take the points 
respectively, 
such that
Show that 
are concurrent.
be the feet of the heights of an acute triangle On the segments take the points respectively, such thatShow that are concurrent.
, we get that 
are concurrent. Then by Cevian Nest Theorem, we have that 
concur.
and 
Prove that
.
Prove that
and 
Prove that
