A sequel to the first six books of the Elements of Euclid, containing an easy introduction to modern geometry, with numerous examples. By John Casey.

172

172 ~A SEQUEL TO EUCLID.

minimum value when the squares which occur on the
right-hand side of the identity vanish; that is, when
a y z
CopR.-K is the miean centre of the, feet of zis own
perpendiculars on the sides of the triangle ABC.
DE, F.-If We put - tan w, w is called the Brocard
a
angle of the triangle.
Prop. 8.-cot co.= cot A, ~ cot B ~ cot C.
Dem.-From 6, ('or. 1, we have
tan o= 42
a ~ 62 + c2'
Hence
co a2~+b2~c 2 _2bc cos A.+2ca cos B +2abcos C
cot W 21=cs
but 4z cs.- cot A, &C.
Hence cot w = cot A + cot B ~ cotOC.
COR.-If the base and the, Brooar~d angle of a triangle
be given, the locus of the symmnedian point is a right line
parallel to the base.
Exercises.
1. if EU, Kb, K, be the points of intersection of the symmedians with the sides of the triangle ABC, the area of the triangle
Kpbi a2+ l~21 + e2)2'