PROPOSITION 12.
To three given straight lines to find a fourth proportional.
Let
A,
B,
C be the three given straight lines; thus it is required to find a fourth proportional to
A,
B,
C.
Let two straight lines
DE,
DF be set out containing any angle
EDF; let
DG be made equal to
A,
GE equal to
B, and further
DH equal to
C; let
GH be joined, and let
EF be drawn through
E parallel to it. [
I. 31]
Since, then,
GH has been drawn parallel to
EF, one of the sides of the triangle
DEF, therefore, as
DG is to
GE, so is
DH to
HF. [
VI. 2]
But
DG is equal to
A,
GE to
B, and
DH to
C; therefore, as
A is to
B, so is
C to
HF.
Therefore to the three given straight lines
A,
B,
C a fourth proportional
HF has been found. Q. E. F.
PROPOSITION 13.
To two given straight lines to find a mean proportional.
Let
AB,
BC be the two given straight lines; thus it is required to find a mean proportional to
AB,
BC.
Let them be placed in a straight line, and let the semicircle
ADC be described on
AC; let
BD be drawn from the point
B at right angles to the straight line
AC, and let
AD,
DC be joined.
Since the angle
ADC is an angle in a semicircle, it is right. [
III. 31]
And, since, in the right-angled triangle
ADC,
DB has been drawn from the right angle perpendicular to the base, therefore
DB is a mean proportional between the segments of the base,
AB,
BC. [
VI. 8, Por.]
Therefore to the two given straight lines
AB,
BC a mean proportional
DB has been found. Q. E. F.
PROPOSITION 14.
In equal and equiangular parallelograms the sides about the equal angles are reciprocally proportional; and equiangular parallelograms in which the sides about the equal angles are reciprocally proportional are equal.
