[/quote]
,
is the midpoint of 
,
the point of intersection of the tangents at points 
of the circumscribed circle of 
is the midpoint of the segment 
.
meets the circumcircle 
.
.
:
be reals such that 
and 
Prove that
Where 
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of 
is the sum of 
,![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.\]](http://latex.artofproblemsolving.com/7/b/6/7b6e32d32d89ca058a06974e7276f0a5874d90a7.png)
such that 
that satisfy the equation
.
.
and 
,
and 
,
is the Euler line of triangle 
.
,
,
,
are odd primes and![\[2^ap^b=(p+2)^c-1.\]](http://latex.artofproblemsolving.com/5/8/5/58518a8ebed14836b72af743ff100eae2efba5e2.png)
we have 
.
.
Y by
An arbitrary triangle ABC with orthocenter 
is given. The inner and external bisectors of angle 
intersect the line 
at points 
and 
, respectively. Two circles are considered: 
the circumcircle of the triangle 
, 
having segment 
as it's diameter.
1. Let point
be such that 
is the angle bisector of the triangle 
. Prove that 
is the external bisector of the same triangle.
2. Let
be a point of intersection of the circles and such that and lie on opposite sides wrt line . Prove that the point lies at the alitude 
of the triangle 
.
3. Let
be a point of intersection of the circles and such that the points and lie on the same side wrt line . Prove that point lies on the median 
.
4. Prove that the tangent to the circle and at the point of intersection with the median intersects the line at the midpoint of the segment .
is given. The inner and external bisectors of angle 
intersect the line 
at points 
and 
, respectively. Two circles are considered: 
the circumcircle of the triangle 
, 
having segment 
as it's diameter.1. Let point
be such that 
is the angle bisector of the triangle 
. Prove that 
is the external bisector of the same triangle.2. Let
be a point of intersection of the circles and such that and lie on opposite sides wrt line . Prove that the point lies at the alitude 
of the triangle 
.3. Let
be a point of intersection of the circles and such that the points and lie on the same side wrt line . Prove that point lies on the median 
.4. Prove that the tangent to the circle
and at the point of intersection with the median intersects the line at the midpoint of the segment .This post has been edited 3 times. Last edited by parmenides51, Mar 4, 2021, 9:51 PM
