Constructing a Line through a Given Point and an Inaccessible Point
Constructing a Line through a Given Point and an Inaccessible Point
Given a point and two lines , that intersect at an inaccessible point , construct a straight line through and .
M
p
q
P
M
P
Let the line through intersect and at and , respectively.
r
M
p
q
A
B
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A
B
M
Let be another point on .
C≠A
p
Let a line parallel to intersect and at and , respectively.
s≠r
r
p
q
C
D
Draw the segments and .
BC
CD
Let be the intersection of with a straight line through that is parallel to .
S
BC
M
p
Let be the intersection of and the straight line through that is parallel to .
N
CD
S
q
The extension of the segment is the desired line.
MN
We have since by the side-splitter theorem.
AM:MB=CN:ND
BS:SC=BM:MA