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Fixing It

A and B are two fixed points on a circle and RS is a variable diamater. What is the locus of the intersection P of AR and BS?

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Set Square

A triangle PQR, right angled at P, slides on a horizontal floor with Q and R in contact with perpendicular walls. What is the locus of P?

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Strange Rectangle

ABCD is a rectangle and P, Q, R and S are moveable points on the edges dividing the edges in certain ratios. Strangely PQRS is always a cyclic quadrilateral and you can find the angles.


Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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.]