You are given a circle of radius 1 unit and two circles of radius $a$ and $b$ which touch each other and also touch the unit circle. Prove that you can always draw a 'flower' with six petals (as in the sketch) with the unit circle in the middle, and six circles around it having radii $a$, $b$, $b/a$, $1/a$, $1/b$ and $a/b$, such that each outer circle touches the unit circle and the two circles on either side of it.

[Note: this diagram is not drawn accurately. Drawing your own more accurate diagram may help you to do the question.]

Mathematical reasoning & proof. Dynamic geometry. Pythagoras' theorem. Regular polygons and circles. Enlargements and scale factors. Interactivities. Circle properties and circle theorems. Angles - points, lines and parallel lines. Similarity and congruence. Golden ratio. 2D shapes and their properties.