Constructions with straightedge and squares only.
We shall consider here the use of squares to draw a perpendicular through a given point to a given line. Full stop.
Obviously, we can then draw parallels (perpendicular to perpendicular), if we accept the use of "auxiliary points",
arbitrary choosen points such that the final construction doesn't depend on the choice of these points.

Midpoint and perpendicular bisector

Draw the two perpendiculars in A and B.
Choosing an arbitrary point (cyan), a perpendicular to these perpendiculars (hence a parallel to AB) makes a rectangle.
The diagonals intersect at the center of this rectangle, and the perpendicular through I to AB
is the perpendicular bisector of AB, hence intersects AB in its midpoint.


To construct A' reflection of A through a given line.
Construct on any perpendiculaire (through arbitrary cyan point) any segment BC and its midpoint M.
A pencil of lines from O, intersect point of AC and MH, allows to complete the construction, for HA/HA' = MC/MB.

Constructible points

These rules don't allow to construct all what can be constructed with compass and straightedge :
for instance we can't copy a segment onto a not parallel line.
Hence we can't construct a square from a given side.

Similitude center

Given points A,B,A',B', (AB not parallel to A'B')
To construct the center of the direct similitude changing AB → A'B'.
Recall the classical construction :
Let I the intersect point of AB and A'B'
The circumcircles to IAA' and IBB' intersect in another point which is the center of the similitude.
(proof from angle property of similitude).
We really can construct this point with straightedge and squares only.
Construct the perpendicular bisectors of IA et IA', intersecting in O,
and also the perpendicular bisectors of IB and IB', intersecting in O'.
We can't draw the circumcircles (centered in O and O'), but the other intersection point is the symmetric S of I with respect to center line OO'.
We already have a segment IA and its midpoint M, just complete the construction of S :
The perpendicular to OO' from I intersects OO' in H.
A parallel to MH from A intersects IH in S.

Well, this construction is unnecessarily complicated.
See below.

False squares

We consider here a false squares, the right angle not being exactly 90░.
The midpoint of a segment can be constructed as above (diagonals of a parallelogram intersects at their midpoints).
But the perpendicular bisector is a little tricky :
Just flip the squares (exchange the role of its sides) to draw angle -θ, hence an isosceles triangle ABO and that's it.
The problem is when θ ≈ 90° for the intersection point of lines at +θ from A and -θ from B is ... very far (at infinity if θ = 90°)
Any parallel to AB makes then an isosceles trapezoid, whose diagonals intersect on the perpendicular bisector.
Drawing two such trapezoids gives two points on the perpendicular bisector.

 We can draw perpendiculars even with a false squares ! 

Center of similitude with a false squares

Sorry, your browser is not Java compliant. This applet allows to choose the angle θ of the squares.
Points A,B,A',B' can be freely moved.

Let I the intersect point of AB and A'B'
Draw lines at angle θ with AB from A and B, and similarily the lines at angle θ with A'B' from A' and B'.
Corresponding lines intersect in a, b. Draw line ab.
Draw a line Is such that angle of ab with Is is θ.
In other words, draw a line from I at angle -θ with ab : (ab, Is) = - (Is, ab)
Is and ab intersect at the searched similitude center S.
(proof with oriented angles of lines and suggested circles in yellow).
With θ = 90°, this gives a much simpler construction than the first one !

It is then quite easy to draw the transformed from any point M by the similitude :
Draw Mq // AB, intersecting IS in Q.
Draw line Mp at angle θ with Mq (parallel to Aa), intersecting ab in P.
Draw line Qm' // A'B'
Construct the line Pm' at angle θ with Qm' (parallel to A'a)
M' is the intersection point of Qm' with Pm'.
Here also, equal angles and concyclic points M,M',P,Q,S is the key.



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