# Help with Ptolemy metric space

#### ypatia

Ptolemy metric space. Help!!

The problem is :
"Let x,y,z,t belongs to R^n where d(x,y)=||x-y||.
Show that(Ptolemy's inequality):
d(x,y)d(z,t)<=d(x,z)d(y,t)+d(x,t)d(y,z)"

I have found this related to the topic paper but I cannot show that the Euclidean space R^n is Ptolemy.
The paper in the second page "2.Preliminaries" says that "To show that the Euclidean space R^n is Ptolemy, consider again four points x,y,z,w. Applying a suitable Mobius transformation we can assume that z is a midpoint of y and w, i.e. |yz|=|zw|=1/2 |yw|. For this configuration...."
But how can we extract from the above paragraph that the Euclidean space R^n is Ptolemy and which is the "suitable Mobius transformation"??

#### fresh_42

Mentor
2018 Award
look at the triangle
with the separate point D on its circumference with radius r and the associated base triangle
, The formula for calculating the side lengths of a base triangle then returns for
:

But now that D is on the perimeter of
is, is
degenerate and its sides lie on the corresponding Simson line , so that the two sides LM and NM complement each other to the third side LN . It therefore applies:

With the above equations, this provides:

If D is not on the circumference, then due to the triangle inequality for
:

The above equations then provide the inequality of Ptolemy:

Source: https://de.wikipedia.org/wiki/Satz_von_Ptolemäus

"Help with Ptolemy metric space"

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