Difference between revisions of "Ptolemy's Inequality"
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| − | with equality for any quadrilateral <math>ABCD</math> with diagonals <math>AC </math> and <math>BD </math>. | + | with equality for any cyclic quadrilateral <math>ABCD</math> with diagonals <math>AC </math> and <math>BD </math>. |
This also holds if <math>A,B,C,D</math> are four points in space not in the same plane, but equality can't be achieved. | This also holds if <math>A,B,C,D</math> are four points in space not in the same plane, but equality can't be achieved. | ||
Revision as of 10:53, 27 February 2020
Ptolemy's Inequality is a famous inequality attributed to the Greek mathematician Ptolemy.
Contents
[hide]Theorem
The inequality states that in for four points 
in the plane,
,
with equality for any cyclic quadrilateral 
with diagonals 
and 
.
This also holds if 
are four points in space not in the same plane, but equality can't be achieved.
Proof for Coplanar Case
We construct a point 
such that the triangles 
are similar and have the same orientation. In particular, this means that
.
But since this is a spiral similarity, we also know that the triangles 
are also similar, which implies that
.
Now, by the triangle inequality, we have 
. Multiplying both sides of the inequality by and using 
and 
gives us
,
which is the desired inequality. Equality holds iff. 
, , and 
are collinear. But since the triangles 
and 
are similar, this would imply that the angles 
and are congruent, i.e., that is a cyclic quadrilateral.
Outline for 3-D Case
Construct a sphere passing through the points 
and intersecting segments 
and 
. We can now prove it through similar triangles, since the intersection of a sphere and a plane is always a circle.
Proof for All Dimensions?
Let any four points be denoted by the vectors 
.
Note that
.
From the Triangle Inequality,
.
Note about Higher Dimensions
Similar to the fact that that there is a line through any two points and a plane through any three points, there is a three-dimensional "solid" or 3-plane through any four points. Thus in an n-dimensional space, one can construct a 3-plane through the four points and the theorem is trivial, assuming the case has already been proven for three dimensions.
