Awkward inversion
by math_explorer, Sep 15, 2010, 1:42 PM
This solution uses an inversion.
and 
are points on circle 
. 
is on the tangent to circle through . 
is the midpoint of 
. 
and 
intersect the circle at resp. 
and 
. If the tangents to circle through and intersect at 
and 
are collinear, prove 
. Hmm, Bulgaria.
The next problem was selected randomly from all posts in the subforum Geometry Unsolved Problems. It looks easy. It probably is, since it's among the 250 oldest posts there. Silly random.org.
46337
are points on a line in that order. A line through intersects the circle with diameter at 
and 
, the circle with diameter at 
, and the circle with diameter 
at 
. Prove that 
= 
. (Note that it doesn't matter if and are swapped.)
maps a point 
such that 
.Quote:
and 
are points on circle 
. 
is on the tangent to circle through . 
is the midpoint of 
. 
and 
intersect the circle at resp. 
and 
. If the tangents to circle through and intersect at 
and 
are collinear, prove 
. Hmm, Bulgaria.
.
are concyclic.
,
.
is the tangent to circle 
.
are concyclic.
are concyclic, and 
is a diameter of that circle. Therefore, 
.
is tangent to circle 
,
,
,
.The next problem was selected randomly from all posts in the subforum Geometry Unsolved Problems. It looks easy. It probably is, since it's among the 250 oldest posts there. Silly random.org.
46337
are points on a line in that order. A line through intersects the circle with diameter at 
and 
, the circle with diameter at 
, and the circle with diameter 
at 
. Prove that 
= 
. (Note that it doesn't matter if and are swapped.)This post has been edited 1 time. Last edited by math_explorer, Sep 15, 2010, 1:42 PM
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