##### Question 10.6.3

Let’s think about cyclic quadrilaterals, that is, quadrilaterals whose four corners lie on a circle. These were of special interest to ancient astronomers such as Ptolemy, studying the angles between stars and planets on the celestial sphere.

1. If we draw in the diagonals $AC$ and $BD$ of the cyclic quadrilateral $ABCD$, can you see any congruent angles? Show hint

2. Can you see any similar triangles? Show answer

3. Find $AC\div BD$ in terms of $AB$, $BC$, $CD$, and $DA$. This gives Ptolemy’s second theorem. You can skip this part if you like. Show hint

4. Maybe we’re not quite satisfied with that, and we’d like some more similar triangles to see what else we can figure out. We might decide to create one. Noting that $\angle ADB = \angle ACB = \delta$, we could create a triangle similar to $\Delta ABC$, by placing $F$ on $BD$ such that $\angle DAF = \angle CAB = \alpha$: This ensures that $\Delta ABC \sim \Delta AFD$. Have we created any other similar triangles? Show hint

5. Use these two similar triangles to find another equation relating the four sides $AB$, $BC$, $CD$, $DA$, and the two diagonals $AC$ and $BD$. This is Ptolemy’s theorem, which will prove very useful in Section 11.4.2: Angle Formulae. Show hint