how to draw a perpendicular bisector

This construction shows how to draw the perpendicular bisector of a given line segment with compass and straightedge or ruler. This both bisects the segment (divides information technology into two equal parts, and is perpendicular to it. It finds the midpoint of the given line segment.

This construction works by effectively building congruent triangles that outcome in right angles being formed at the midpoint of the line segment. The proof is surprisingly long for such a simple construction.

The image below is the final drawing to a higher place with the red lines and dots added to some angles.

	Statement	Reason
1	Line segments AP, AQ, PB, QB are all congruent	The 4 distances were all fatigued with the aforementioned compass width c.
Next we prove that the top and bottom triangles are isosceles and congruent
2	Triangles ∆APQ and ∆BPQ are isosceles	Two sides are coinciding (length c)
3	Angles AQJ, APJ are congruent	Base of operations angles of isosceles triangles are congruent
4	Triangles ∆APQ and ∆BPQ are coinciding	Three sides congruent (sss). PQ is mutual to both.
5	Angles APJ, BPJ, AQJ, BQJ are coinciding. (The iv angles at P and Q with red dots)	CPCTC. Corresponding parts of congruent triangles are congruent
Then we prove that the left and right triangles are isosceles and congruent
6	∆APB and ∆AQB are isosceles	Two sides are congruent (length c)
7	Angles QAJ, QBJ are coinciding.	Base of operations angles of isosceles triangles are coinciding
eight	Triangles ∆APB and ∆AQB are congruent	Iii sides coinciding (sss). AB is common to both.
ix	Angles PAJ, PBJ, QAJ, QBJ are coinciding. (The four angles at A and B with bluish dots)	CPCTC. Corresponding parts of congruent triangles are coinciding
Then we prove that the four pocket-sized triangles are congruent and finish the proof
10	Triangles ∆APJ, ∆BPJ, ∆AQJ, ∆BQJ are congruent	2 angles and included side (ASA)
eleven	The iv angles at J - AJP, AJQ, BJP, BJQ are congruent	CPCTC. Corresponding parts of congruent triangles are coinciding
12	Each of the iv angles at J are 90°. Therefore AB is perpendicular to PQ	They are equal in measure and add to 360°
xiii	Line segments PJ and QJ are congruent. Therefore AB bisects PQ.	From (eight), CPCTC. Respective parts of congruent triangles are congruent