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A circle is inscribed in an equilateral triangle. Smaller circles touch it and the sides of the triangle, the process continuing indefinitely. What is the sum of the areas of all the circles?

### Areas and Ratios

Do you have enough information to work out the area of the shaded quadrilateral?

### Climbing Powers

$2\wedge 3\wedge 4$ could be $(2^3)^4$ or $2^{(3^4)}$. Does it make any difference? For both definitions, which is bigger: $r\wedge r\wedge r\wedge r\dots$ where the powers of $r$ go on for ever, or $(r^r)^r$, where $r$ is $\sqrt{2}$?

# Impossible Square?

### Why do this problem

This question is a nice introduction to the concept of proof by contradiction: something concrete (the area of a square) is calculated in two different ways, and these ways are shown to be inconsistent.

### Possible approach

This problem could be offered with no advice. Students may experiment with the ideas of visual proofs before realising that they will need to look at an algebraic solution.

Once the algebra has led to the solution, it is a useful exercise to ask students to explain in words why the absurdity leads to the rejection of the square shape, rather than the rejection of one of the methods of calculation or the rejection of some of their assumptions concerning the base triangles or the concept of area.

If the students believe that the square shape is impossible, they could be pressed on why they are so sure of their methods of calculation. Such discussion may lead to an increased appreciation of the need for axioms in mathematics which are statements that are agreed by all to be true and from which all else follows.

### Key Questions

What formulas for area will we need to use in this question?
How can we relate these two formulas?

[once someone claims to have solved the problem] Can you explain to the class why we cannot make a square?

### Possible extensions

For the interested student, the article Proof by Contradiction offers interesting and stimulating reading on the concept of proof by contradiction.

Alternatively, students could try to experiment to find other shapes which cannot be made from the triangles. For example,
• Which rectangles is it possible to make using these triangles?
• How many orientations of triangle are possible in a general completed jigsaw?

### Possible support

To get started, why not try the simpler related problem The Square Hole