Multiplication in Euclidean Geometry

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In this exercise you are given four points \(P\), \(Q\), \(R\) and \(S\) on a line such that \(|PQ| = 1\). Your task is to construct a point \(T\) on this line such that \(|PT| = |PR| \times |PS|\). Once you have constructed \(T\) you can use the length measurement tool to confirm that \(|PT|\) is the correct length.

Hint. You will find Thales' theorem very useful!

The diagram contains an example of four points \(A\), \(B\), \(C\), \(D\) and \(E\) on a line such that \(|AB| = 1\) and \(|AE| = |AC| \times |AD|\).


Remark. You have probably noticed that this construction is relative to the length \(|AB|\). The distances in the picture are measured in centimetres, but we could just as well declare the length \(|AB|\) equal to "1 unit" and measure all the other distances relative to this. It makes sense that we need to define "1 unit" to do this, since "multiplication by one" has a precise meaning.

Challenge. There are two different ways to construct the point \(T\). Can you use one of the theorems from class to prove that these both give the same result? This is the geometric version of commutativity of multiplication of real numbers \(|PR| \times |PS| = |PS| \times |PR|\).

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