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
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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
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.