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Start with any triangle T1 and its inscribed circle. Draw the triangle T2 which has its vertices at the points of contact between the triangle T1 and its incircle. Now keep repeating this process starting with T2 to form a sequence of nested triangles and circles. What happens to the triangles? You may like to investigate this interactively on the computer or by drawing with ruler and compasses. If the angles in the first triangle are a, b and c prove that the angles in the second triangle are given (in degrees) by f(x) = (90 - x/2) where x takes the values a, b and c. Choose some triangles, investigate this iteration numerically and try to give reasons for what happens. Investigate what happens if you reverse this process (triangle to circumcircle to triangle...)

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# Vedic Sutra - All from 9 and Last from 10

##### Stage: 4 Challenge Level:

Well done Geoffrey, University College School, London for this excellent solution.

This problem is actually remarkably straightforward. A little bit of algebra is all that is needed.

Let x and y each denote 1 of the 2 numbers to be multiplied together.

The layout of the grids in the question then corresponds to:

 x ( 10 - x ) y ( 10 - y )

The 4 ways of calculating the left hand digit are expressed algebraically as follows:

1. x + y - 10
2. 10 - [ ( 10 - x ) + ( 10 - y ) ] = 10 - ( 20 - x - y ) = x + y - 10
3. x - ( 10 - y ) = x + y - 10
4. y - ( 10 - x ) = x + y - 10

All 4 ways are algebraically equivalent, so this is why they give the same answer.

To explain why the Vedic Sutra works, we run through the whole method algebraically.

The right hand digit is ( 10 - x ) * ( 10 - y ).
The left hand digit is ( x + y - 10 ). We should remember that this is the tens digit and multiply by 10 before summing both sides.

10 * ( x + y - 10 ) + ( 10 - x ) * ( 10 - y )
= 10x + 10y - 100 + 100 - 10x - 10y + xy
= xy

Using the Vedic Sutra, we get the result xy, which is indeed the product of x and y.