Plane and solid analytic geometry, by William F. Osgood and William C. Graustein.

THE ELLIPSE

109

instead of the flexible string we had a smooth rigid wire, in the
form of the ellipse, for the bead to slide on, the bead would be
held at P by the cord exactly as before. But the reaction of
a smooth wire is at right angles to its tangent. This is the
very conception of a smooth wire. For otherwise, if S were
oblique, it could be resolved into a normal and a tangential
component. But the smooth wire could not yield a reaction,
part of which is along the tangent.
It follows, then, that the normal at P bisects the angle between the focal radii, and hence these make equal angles with
the tangent at P, q. e. d.
ii) Proof by Means of Minimum Distances. A Lemma. A
barnyard is bounded on one side by a straight river. The
cows, as they come from the pasture,
enter the barnyard by a gate at A, A B
go to the river to drink, and then \
keep on to the door of the barn at B. L i pC
What point, P, of the river should a '-.,
cow select, in order to save her steps FIG. 7
so far as possible?
It is easy to answer this question by means of a simple construction. From B drop a perpendicular BM on the line of
the river bank, L, and produce it to B', making MB' = BM.
Join A with B' and let AB' cut L at C. Then C is the position of P, for which the distance under consideration,
AP + PB,
is least.
For, the straight line AB' is shorter than any broken line
APB':
AB'< APB'.
But PB = PB' and CB = CB'.
Hence
AB'= AC+ CB and APB'= AP+ PB.
It follows, then, that
AC + CB < AP + PB,