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Today Puzzle #645

Puzzle No. 635 – Thursday 2 January 2020

Pythagoras has drawn a three-four-five triangle. In it he inscribes a circle that just touches the three sides. What is the radius of the circle?

Today’s #PuzzleForToday has been set by Hugh Hunt, Reader in Engineering Dynamics and Vibration at Trinity College, Cambridge

Click here for the answer


AB is the hypotenuse of triangle ABC with sides a (opposite A), b opposite

(B) and c. The circle of radius r, centred at O, touches the sides at D on

AB, E on BC and F on CA.

Given C is a right angle, OECF is a square of side r.

OEB and ODB are both right angles so DOEB is a kite with sides r and a-r

and so we deduce that BD is of length a-r and AD is of length c-a+r.

Likewise AFOD is a kite of sides r and b-r from which we deduce that c-a+r

= b-r

or 2r = a+b-c [eq.1]

And with a,b,c = 3,4,5 we get r=1

It is interesting to note that for all Pythagorean triples (integer values)

the radius is always integer. This is because from a^2 + b^2 = c^2 we get

that either all of a, b and c are even or two of them are odd. This means

from [eq.1] that 2r is even, so r is always an integer.

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