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.

14

A SEQUEL TO EUCLID.

Prop. 19.-Through a given point P to draw a line
which shall form, with two given lines CA, CB, a triangle
of minimum area. c
Con.-ThroughP draw
PD || to CB; cut off AD D
= CD; join AP, and produce to B. Then AB is A
the line required. BDem.-Let RQ be any \
other line through P;
draw AM 1| to CB. Now, because AD = DC, we
have AP = P:B; and the As APM and QPB have the
Z s APM, AMP respectively equal to BPQ, BQP, and
the sides AP and PB equal to one another;.'. the
triangles are equal; hence the A APR is greater than
BPQ: to each add the quadrilateral CAPQ, and we get
the A CQR greater than ABC.
Cor. 1.-The line through the point P which cuts
off the minimum triangle is bisected in that point.
Cor. 2.-If through the mid- A
dle point P, and through any
other point D of the side AB of R
the A ABC we draw lines 1I to p
the remaining sides, so as to -I -
form two inscribed s s CP, CD,
then CP is greater than CD.
Dem.-Through D draw c
QR, so as to be bisected in D; B q
then the A ABC is greater than CQR; but the Us
are halves of the As; hence CP is greater than CD.
A very simple proof of this Cor. can also be given by
means of (xliii.)
Prop. 20.- When two sides of a triangle are given in
magnitude, the area is a maximum when they contain a
right angle.
Dem.-Let BAC be a A having the Z A right;
with A as centre and AC as radius, describe a;
take any other point D in the circumference; it is