[/quote]
,
.
intersects 
at 
,
at 
respectively. Suppose 
and 
intersect again at 
,
is the angle bisector of 
.
and 
Prove that
Let 
Prove that
be complex numbers satisfying 
.
complex numbers such that the modulus of the sum of these complex numbers is not less than 
.
and 
be three positive integers. Prove that there exist a positive integer 
and a set of 
positive integers 
,
, the 
-
-
-
.
, s.t :
be an acute triangle with 
,
of side 
)
.
and 
,
,
.
intersects line 
at a second point 
.
and 
are perpendicular.
satisfy,
Where 
}
has a unique solution in integers with 
and 
.
as 
fir 
.
for all 
.
and orthocenter 
.
intersect 
respectively. Lines through 
intersects 
at 
is a parallelogram. Prove that lines 
and 
.
is a cyclic quadrilateral inscribed in a circle of radius one unit. If 
,
are positive real numbers.Prove that
Y by
Let 
be an isosceles triangle with the base 
, the point 
lies on the side 
. After rotating with center at point , point 
moves at point 
, which lies on the extension of side 
beyond point 
, and point moves at point 
in the midpoint of side . Using a compass and a ruler, restore the triangle , if only the points and are given.
be an isosceles triangle with the base 
, the point 
lies on the side 
. After rotating with center at point , point 
moves at point 
, which lies on the extension of side 
beyond point 
, and point moves at point 
in the midpoint of side . Using a compass and a ruler, restore the triangle , if only the points and are given.This post has been edited 3 times. Last edited by parmenides51, Dec 20, 2022, 1:41 PM
