The Enjoyment of Elementary Geometry Li Zhou ... - MAA Sections

The Most Important Fact about the Brocard Angle =
cot= œ cotE cotF cotG
An Angle of Many Inequalities
For example,
(a) = Ÿ$! ° (equality iff EFGis equilateral);
(b) = $ Ÿ ÐE = ÑÐF = ÑÐG = Ñ;
(c)
# = Ÿ È $
EFG
(d) # Ÿ .
$
=
" " "
E F G
(the Yff inequality);
An IMO Problem
IMO Problem 1991/5. Let EFG be a triangle and T an interior point in EFG . Show that at least
one angles nT EF, nT FGß nT GE is less than or equal to $! °.
Solution. T must be in one of the three smaller triangles HEF, HFG, or H GE. Since = Ÿ $! °,
the proof is complete.
A Recent MONTHLY Problem
A recent MONTHLY problem offers a new inequality of the Brocard angle.
Problem 11017 [2003, 439]. Proposed by C. R. Pranesachar, Indian Institute of Science, India.
Let T and Ube the Brocard points of a non-equilateral triangle X , and let = be the Brocard angle
of X .
(a) Prove that the line through T and U passes through a vertex of X if and only if the sides of the
triangle are in geometric progression.
1 FG
' $
(b) Prove that = min Ò ß Ó, where F and G are the smallest angles of X.