Difference between revisions of "Ptolemy's Inequality"
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which is the desired inequality. Equality holds iff. <math>A </math>, <math>P </math>, and <math>{C} </math> are [[collinear]]. But since the angles <math>BAP </math> and <math>BDC </math> are congruent, this would imply that the angles <math>BAC </math> and <math>BPC </math> are [[congruent]], i.e., that <math>ABCD </math> is a cyclic quadrilateral. | which is the desired inequality. Equality holds iff. <math>A </math>, <math>P </math>, and <math>{C} </math> are [[collinear]]. But since the angles <math>BAP </math> and <math>BDC </math> are congruent, this would imply that the angles <math>BAC </math> and <math>BPC </math> are [[congruent]], i.e., that <math>ABCD </math> is a cyclic quadrilateral. | ||
+ | [[Category:Geometry]] | ||
+ | [[Category:Inequalities]] | ||
[[Category:Theorems]] | [[Category:Theorems]] |
Revision as of 14:46, 8 November 2007
Ptolemy's Inequality states that in for four points in the plane,
,
with equality iff. is a cyclic quadrilateral with diagonals
and
.
Proof
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 angles
and
are congruent, this would imply that the angles
and
are congruent, i.e., that
is a cyclic quadrilateral.