## 1. The Law of Sines

For this definition to make sense, we need the

Proof. Let `A` be the center of a circle. First we consider the case where one side is a diameter. Let `DB` be a diameter. Then for a point `C` on the circle, `/_BAC` is an exterior angle of the isoceles △`DAC`.

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By the Exterior Angle Theorem of Euclid, we have

So the peripheral angle is half the central angle to the same secant `CB`.

Recall the Law of Sines from trigonometry:

## 2. The Cross Ratio

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Proof of `(**)`. Apply the Law of Sines to △`ABP` then

 `frac{A-B}{sin/_BPA}=frac{a}{sin/_B}=frac{b}{sin/_A}`,

where `A-B` is the signed length `|A-B|` and `sin/_B=sin/_PBA=sin/_CBP` since supplementary angles have the same sine. Now let’s calculate

 `frac{A-B}{A-D}\ frac{C-D}{C-B}\ =\ frac{A-B}{sin/_BPA}\ frac{C-D}{sin/_DPC}\ =\ frac{P-A}{sin/_B}\ frac{P-C}{sin/_D}`.

Similarly,

 `frac{sin/_BPA}{sin/_DPA}\ frac{sin/_DPC}{sin/_BPC}\ =\ frac{A-D}{sin/_DPA}\ frac{C-B}{sin/_BPC}\ =\ frac{P-A}{sin/_D}\ frac{P-C}{sin/_B}`.

So `frac{A-B}{A-D}\ frac{C-D}{C-B}\ =\ frac{sin/_BPA}{sin/_DPA}\ frac{sin/_DPC}{sin/_BPC}\ \ square`.

## 3. Perspective Rulers

A perspective ruler on a line with a finite vanishing point is obtained with the following procedure: given `l` with vanishing point `V`,

• Choose a point `P` not on `l` on the extension of `l` past `V`.

• Choose a line `s\ ||\ (PV)`.

• On `s` choose a Euclidean ruler.

• Transfer this scale through the lens `P` to `l`.

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