Symmetry
A proof utilizes symmetry if the steps to prove one thing is identical to those steps of another. For example, to prove that in triangle ABC with all three sides congruent to each other that all three angles are equal, you only need to prove that if then
the other cases hold by symmetry because the steps are the same.
Hidden symmetry
Let the convex quadrilateral be given.
Prove that
Proof
Let be bisector
Let point be symmetric
with respect
is isosceles.
Therefore
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Composition of symmetries
Let the inscribed convex hexagon be given,
Prove that
Proof
Denote the circumcenter of
the common bisector
the common bisector
the smaller angle between lines
and
the symmetry with respect axis
the symmetry with respect axis
It is known that the composition of two axial symmetries with non-parallel axes is a rotation centered at point of intersection of the axes at twice the angle from the axis of the first symmetry to the axis of the second symmetry.
Therefore
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