Let acute-angled triangle
, and let
be the circumscribed circle. Points
and
in the parts
and
corresponding order
.
The perpendicular bisector of
and
intersecting the small arches
and
the
in points
and
respectively.
Prove that the lines
and
are either parallel or are identical.
Suppose that
intersects the circumscribed circle of
a point
.
We observe that
.
Thus the triangle
is isosceles with
.
Similarly if
intersects circumcircle of
at
, then
.
We will prove that
, ie
.
We note that
, while
(
is the core of
the circle with center
and radius
.
So it suffices to prove that
, that
.
But we know that
, by definition
.
So long
.
We observe that the quadrilateral
is recordable and
therefore the
is the bisector
.
If it
was not equilateral, then since
it belongs to the bisector
and
, should it
be writable (south pole theorem) inappropriate, since the circumscribed circle
is the circumscribed circle
and
does not belong to this circle.
So it
is isosceles with
and the requested follow.







The perpendicular bisector of







Prove that the lines


Solution
Suppose that



We observe that

Thus the triangle


Similarly if




We will prove that


We note that






So it suffices to prove that


But we know that


So long

We observe that the quadrilateral




If it








So it


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