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Three Balls

A circle has centre O and angle POR = angle QOR. Construct tangents at P and Q meeting at T. Draw a circle with diameter OT. Do P and Q lie inside, or on, or outside this circle?

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The centre of the larger circle is at the midpoint of one side of an equilateral triangle and the circle touches the other two sides of the triangle. A smaller circle touches the larger circle and two sides of the triangle. If the small circle has radius 1 unit find the radius of the larger circle.

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The Eyeball Theorem

Two tangents are drawn to the other circle from the centres of a pair of circles. What can you say about the chords cut off by these tangents. Be patient - this problem may be slow to load.

Some(?) of the Parts

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Jeremy from Nottingham High School was the first to crack this one with the neatest proof that the diameter of the circle is equal to the perimeter of the triangle $OAB$ and Peter from Konstanz sent a similar proof. Other good proofs came in from Hannah and Sarah of St Philomena's School, Carshalton, Steven Cunnane of Norwich School, and James of The Robert Smyth School, Market Harborough.

If you take $OA$ as 1 unit then the perimeter $OAB$ and the diameter of the circle are equal to $2 + \sqrt{2}$ units.

Let the circle touch $OA$, $OB$ and $AB$ at $X$, $Y$ and $Z$ respectively.

Let $C$ be the centre of the circle and $R$ the radius. It is easy to prove that $OYCX$ is a square.

Then $AZ = AX$, since the tangents from a point to a circle are equal. Similarly, $BZ = BY$.

The perimeter of the triangle is
$OA + OB + AB$
$= OA + OB + AZ + BZ$
$= OA + OB + AX + BY$
$= OX + OY$
$= 2R$.