# 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