be a sequence of positive real numbers that satisfies 
and 
for every natural number 
.
for every natural number 
and 
.
be positive real numbers such that : 
. Prove that : 
.
and 
to 
respectively.
and 
as the reflections of 
be the circumcircle of 
.
with 
with 
is the circumcenter of 
can be inscribed within a circle.
Y by
Let 
be the center of the inscribed circle 
of the triangle 
. The circumscribed circle of the triangle 
intersects at points 
and 
so that and 
lie on one side of the straight line 
, and and 
on the other. We denote by 
the midpoint of the smaller arc 
of the circumscribed circle of the triangle , and by 
the midpoint of the smaller arc 
.
1. Prove that if
, then triangle is isosceles.
2. Given a triangle
. The circle passing through the vertices 
and 
intersects the sides 
and 
at points 
and 
, respectively. The bisector of angle 
intersects at point 
, and the bisector of angle 
intersects at point 
. Prove that 
.
3. Prove that
4. Let
be the point of intersection of lines 
and 
be the point of intersection of lines 
and 
. Prove that 
.
5. Prove that
.
6. Let
be the point of intersection of lines and , and 
be the point of intersection of lines 
and 
. Prove that 
and are collinear.
Grade 9: 1,3-5 Grade 10: 1-2,5-6
be the center of the inscribed circle 
of the triangle 
. The circumscribed circle of the triangle 
intersects at points 
and 
so that and 
lie on one side of the straight line 
, and and 
on the other. We denote by 
the midpoint of the smaller arc 
of the circumscribed circle of the triangle , and by 
the midpoint of the smaller arc 
.1. Prove that if
, then triangle is isosceles.2. Given a triangle
. The circle passing through the vertices 
and 
intersects the sides 
and 
at points 
and 
, respectively. The bisector of angle 
intersects at point 
, and the bisector of angle 
intersects at point 
. Prove that 
.3. Prove that
4. Let
be the point of intersection of lines 
and 
be the point of intersection of lines 
and 
. Prove that 
.5. Prove that
.6. Let
be the point of intersection of lines and , and 
be the point of intersection of lines 
and 
. Prove that 
and are collinear.Grade 9: 1,3-5 Grade 10: 1-2,5-6
This post has been edited 2 times. Last edited by parmenides51, Mar 4, 2021, 9:24 PM
