Advanced High-School Mathematics

SECTION 3.2 Classical Inequalities 153

(iv) Show that this implies thatP(x,y)≥0 whenx, y >0 with

equality if and only ifx=y.)

12. You are given 4 ABC and an in-
    terior pointP with distances x to
    [BC], y to [AC] and z to [AB] as
    indicated. Leta = BC, b = AC,
    andc=AB.
    (a) Find the point P which mini-
    mizes the objective function

F =

a

x

+

b

y

+

c

z

.

x

y

z P

C

B

A

(Hint: note thatax+by+czis proportional to the area of 4 ABC.

If need be, refer back to Exercise 5 on page 17.^4 )

(b) Conclude from part (a) that the inradiusrof 4 ABC(see page

17) is given byr = 2A/P, where AandP are the area and

perimeter, respectively, of 4 ABC.

The next few exercises will introduce a geometrical notion of the mean
of two positive numbers. To do this, fix a positive numbern 6 = 1 (which
need not be an integer), and draw the graph ofy=xnfor non-negative
x. For positive real numbersa 6 =b, locate the pointsP =P(a,an) and
Q=Q(b,bn) on the graph. Draw the tangents to the graph at these
points; thex-coordinate of the point of intersection of these tangents
shall be denotedSn(a,b) and can be regarded as a type of mean ofa
andb. (Ifa=b, setSn(a,b) =a.) See the figure below:

(^4) It turns out thatPmust be the incenter of 4 ABC.