Rays in a Right Triangle.

Rays in a Right Triangle. Peter G. Brown School of Mathematics and Statistics University of New South Wales Sydney, Australia

Stewart’s Theorem is well-known but little used. Here is a very simple application which yields a rather pretty formula for the sum of the squares of certain lengths in a right triangle. Stewart’s Theorem: In the diagram, we have b2 m + c2 n = a(d2 + mn), where a = m + n.

b

d

c

n m Now take a right triangle ABC as shown and place n − 1 equally spaced points along the side AB. Join each of these points to C to form line segments of lengths d1 , d2, ..., dn−1 .

1

A

D c

b = dn di d2

d1 B C a = d0 Consider the segment di and apply Stewart’s Theorem to triangle ABC with CD as the line segment, then we have c ci c2 a2 ( (n − i)) + b2 = c(d2i + 2 i(n − i)). n n n Re-arrange and sum from i = 0 to n we have n X

d2i

=

n X

i=0

i=0

n−i c2 2 i a( ) + b − 2 i(n − i). n n n 2

Summing the right-hand side and using a2 + b2 = c2 we have n X i=0

d2i =

c2 n(n + 1)(2n + 1), 6n2

whence we can write d20 + d21 + ... + d2n =

c2 2 (1 + 22 + ... + n2 ). n2

We finally note that as n gets large, the average of the sum of the squares of these distances approaches 31 the square of the hypotenuse.

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