[/quote]
,
,![\[
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be a parallelogram such that 
Let 
and 
be the midpoints of 
and 
respectively. Assuming that 
is a cyclic quadrilateral, find 
be real numbers such that 
Prove that
Let 
Prove that
be a polynomial with integer coefficients. Define 
and 
for 
.
such that 
,
or 
.
have 
distinct real roots, which sum up to 
.
,
and their corresponding roots.
be a positive integer greater than 
satisfying:
for all integers 
be a triangle, and let 
be an arbitrary point on line 
,
is the circumcenter of 
and 
as the orthocenters of triangles 
and 
.
passes through a fixed point which is independent of the choice of 
be a triangle inscribed in a circle, where the tangents to the circle at points 
be a point on the arc 
(not containing 
and 
.
intersect at point 
be the reflection of 
and 
intersect at point 
,
.
.
,
be the orthocenter of 
.
at 
.
be the midpoint of 
,
meet 
.
at 
.
is tangent to 
.
,
and 
,
is the intersection point of the diagonals 
.
intersects 
and 
.
is tangent to the circles with diameters 
and 
.
such that 
and 
for any real numbers 
and 
.
Y by
The triangle 
is isosceles and has a right angle at the vertex 
. Construct all points that simultaneously satisfy the following two conditions:
(i) are equidistant from points and 
(ii) heve distance exactly three times from point
as far as from point .
is isosceles and has a right angle at the vertex 
. Construct all points that simultaneously satisfy the following two conditions:(i) are equidistant from points
and 
(ii) heve distance exactly three times from point
as far as from point .This post has been edited 1 time. Last edited by parmenides51, Dec 15, 2020, 4:35 AM
is the perpendicular bisector of 
is a circle of apollonius with center on line 
away from 
away from 
.
