Ptolemy's Inequality: Difference between revisions
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with equality if and only if <math>ABCD </math> is a [[cyclic quadrilateral]] with [[diagonal]]s <math>AC </math> and <math>BD </math>. | with equality if and only if <math>ABCD </math> is a [[cyclic quadrilateral]] with [[diagonal]]s <math>AC </math> and <math>BD </math>. | ||
== Proof == | 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. | ||
== Proof for Coplanar Case== | |||
We construct a point <math>P </math> such that the [[triangles]] <math>APB, \; DCB </math> are [[similar]] and have the same [[orientation]]. In particular, this means that | We construct a point <math>P </math> such that the [[triangles]] <math>APB, \; DCB </math> are [[similar]] and have the same [[orientation]]. In particular, this means that | ||
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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>BDC </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>BDC </math> are [[congruent]], i.e., that <math>ABCD </math> is a cyclic quadrilateral. | ||
==Proof for 3-D Case== | |||
Construct a sphere passing through the points <math>B,C,D</math> and intersecting segments <math>AB,AC,AD</math> and <math>E,F,G</math>. We can now prove it through similar triangles, since the intersection of a sphere and a plane is always a circle. | |||
==General Proof== | |||
Let any four points be denoted by <math>\bold a,\bold b,\bold c,\bold d</math>. | |||
Note that | |||
<math>(\bold a-\bold b)\cdot(\bold c-\bold d)+(\bold a-\bold d)\cdot(\bold b-\bold c)</math> | |||
<math>=\bold a\cdot\bold c-\bold a\cdot\bold d-\bold b\cdot\bold c+\bold b\cdot\bold d+\bold a\cdot\bold b-\bold a\cdot\bold c-\bold d\cdot\bold b+\bold d\cdot\bold c</math> | |||
<math>=\bold a\cdot\bold b-\bold a\cdot\bold d-\bold b\cdot\bold c+\bold c\cdot\bold d</math> | |||
<math>=(\bold a-\bold c)\cdot(\bold b-\bold d)</math>. | |||
From the Triangle Inequality, | |||
<math>|(\bold a-\bold b)\cdot(\bold c-\bold d)|+|(\bold a-\bold d)\cdot(\bold b-\bold c)|\ge|(\bold a-\bold c)\cdot(\bold b-\bold d)|</math> | |||
<math>\implies|\bold a-\bold b| |\bold c-\bold d|+|\bold a-\bold d| |\bold b-\bold c|\ge|\bold a-\bold c| |\bold b-\bold d|</math> | |||
<math>\implies AB\cdot CD+AD\cdot BC\ge AC\cdot BD</math>, | |||
with equality when <math>A,B,C,D</math> are on a circle or a line. | |||
==See Also== | ==See Also== | ||
Revision as of 08:31, 11 June 2009
Ptolemy's Inequality is a famous inequality attributed to the Greek mathematician Ptolemy.
Theorem
The inequality states that in for four points 
in the plane,
,
with equality if and only if 
is a 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 angles 
and 
are congruent, this would imply that the angles 
and are congruent, i.e., that is a cyclic quadrilateral.
Proof 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.
General Proof
Let any four points be denoted by 
.
Note that
.
From the Triangle Inequality,
,
with equality when are on a circle or a line.
