Difference between revisions of "Ptolemy's Inequality"
(reworded statement in a more general way) |
(added equality case) |
||
(19 intermediate revisions by 13 users not shown) | |||
Line 1: | Line 1: | ||
− | '''Ptolemy's Inequality''' states that in for four points <math> | + | '''Ptolemy's Inequality''' is a famous inequality attributed to the Greek mathematician Ptolemy. |
+ | |||
+ | ==Theorem== | ||
+ | The inequality states that in for four points <math>A, B, C, D </math> in the plane, | ||
<center> | <center> | ||
− | <math> | + | <math>AB \cdot CD + BC \cdot DA \ge AC \cdot BD</math>, |
− | |||
− | </math>, | ||
</center> | </center> | ||
− | with equality | + | with equality if and only if <math>ABCD</math> is a cyclic quadrilateral with diagonals <math>AC </math> and <math>BD </math> OR if <math>A, B, C, D</math> are collinear. |
+ | |||
+ | 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 == | + | == Proof for Coplanar Case== |
− | We construct a point <math> | + | 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 |
<center> | <center> | ||
<math> | <math> | ||
− | BD = \frac{BA \cdot DC }{AP} \; ( | + | BD = \frac{BA \cdot DC }{AP} \; (1) |
</math>. | </math>. | ||
</center> | </center> | ||
− | But since this is a [[spiral similarity]], we also know that the triangles <math> | + | But since this is a [[spiral similarity]], we also know that the triangles <math>ABD, \; PBC </math> are also similar, which implies that |
<center> | <center> | ||
<math> | <math> | ||
− | BD = \frac{BC \cdot AD}{PC} \; ( | + | BD = \frac{BC \cdot AD}{PC} \; (2) |
</math>. | </math>. | ||
</center> | </center> | ||
− | Now, by the [[triangle inequality]], we have <math> | + | Now, by the [[triangle inequality]], we have <math>AP + PC \ge AC </math>. Multiplying both sides of the inequality by <math>BD</math> and using equations <math>(1) </math> and <math>(2) </math> gives us |
<center> | <center> | ||
<math> | <math> | ||
− | BA \cdot DC + BC \cdot AD \ge AC \cdot | + | BA \cdot DC + BC \cdot AD \ge AC \cdot BD |
</math>, | </math>, | ||
</center> | </center> | ||
− | which is the desired inequality. Equality holds iff. <math> | + | which is the desired inequality. Equality holds iff. <math>A </math>, <math>P </math>, and <math>{C} </math> are [[collinear]]. But since the triangles <math>BAP </math> and <math>BDC </math> are similar, 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. |
+ | |||
+ | ==Outline 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. | ||
+ | |||
+ | ==Proof for All Dimensions?== | ||
+ | |||
+ | Let any four points be denoted by the vectors <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>. | ||
+ | |||
+ | ==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. | ||
+ | |||
+ | ==See Also== | ||
+ | *[[Ptolemy's Theorem]] | ||
+ | |||
+ | [[Category:Geometry]] | ||
+ | [[Category:Inequalities]] | ||
+ | [[Category:Geometric Inequalities]] |
Latest revision as of 08:01, 7 June 2023
Ptolemy's Inequality is a famous inequality attributed to the Greek mathematician Ptolemy.
Contents
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 OR if are collinear.
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 equations 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.