c C'F'3.1 SYMMETRY 67FA tactic that often works for geometric inequalities is to look for a way to compare
the sum of several lengths with a single length, since the shortest distance between
two points is a straight line. Quadrilaterals AGBC' and H EF' D are cyclic, since the
opposite angles add up to 180 degrees. Ptolemy's Theorem then implies that AG ·
BC' + GB. AC' = C' G. AB. Since ABC' is equilateral, this implies that AG + GB = C' G.
Similarly, DH + HE = H F'. The shortest path between two points is a straight line. It
follows that
CF =C'F' � C'G+GH +HF' =AG+GB+GH +DH +HE,
with equality if and only if G and H both lie on C' F'. •
Algebraic Symmetry
Don't restrict your notions of symmetry to physical or geometric objects. For example,
sequences can have symmetry, like this row of Pascal's Triangle:
1 ,6, 15,20,20, 15,6, 1.
That's only the beginning. In just about any situation where you can imagine "pairing"
things up, you can think about symmetry. And thinking about symmetry almost always
pays off.
The Gaussian Pairing Tool
Carl Friedrich Gauss (1777-1855) was certainly one of the greatest mathematicians of
all times. Many stories celebrate his precocity and prodigious mental power. No one
knows how true these stories are, because many of them are attributable only to Gauss
himself. The following anecdote has many variants. We choose one of the simplest.