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# Pentakite

**Can you show that triangle $BFC$ is congrugent to triangle $DAC$?**
**Can you find a pair of similar triangles?**
**Can you use these results to find the length $DA$?**
*We are very grateful to the Heilbronn Institute for Mathematical Research for their generous support for the development of this resource.*
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### At a Glance

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Age 14 to 18

Challenge Level

$ABCDE$ is a regular pentagon of side length one unit.

The lines $AB$ and $DC$ are extended until they meet at another point, $F$.

*You can read more about the Golden Ratio in this Plus Article.*

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

Three triangles ABC, CBD and ABD (where D is a point on AC) are all isosceles. Find all the angles. Prove that the ratio of AB to BC is equal to the golden ratio.