20-August-2008

Construction of a Right Bisector of a Line Segment AB

Step 1.  Draw a circle with diameter 2r > the length of line segment AB about one of the end points:

Step 2: Draw  a second circle of same radius r about the other end point.

Step 3. Label the points of intersection of the two circles with letters P and Q.

Step 4.  Draw the line through the intersection points PQ or the line segment PQ only.

Step 5. Label the intersection of the line segment PQ and line segments AB with a letter D.

The construction is complete. 

Claim: The point D bisects the line segment AB and the line PQ is perpendicular to the line segment AB.  It or the line that extends provides a or the right bisector of the line segment AB.

Proof of the Claim:  Draw triangles PAB and QAB. These triangles are isoceles as both P and Q are at distance r from both A and B.

Moreover, triangles PAB and QAB are isometric by the SSS criteria. Therefore the four angles at A and B in triangles PAB and QAB are all equal:

Next, triangles APQ and BPQ are isometric by the SSS criteria, and they are also isosceles since adjacent sides at A and B in both are equal in length to the radius r. Therefore the four angles at P and Q in these triangles are equal.

Finally triangles AQD, APD, BPD and BQD are isometric by the ASA criteria. The latter isometry implies the four equal angles at D sum to 360 degrees (4 rights angles), and hence each must be a right angle . The latter isometry implies line segment AD and BD are equal in length. So the point D bisect the line segement AD.  And as a after thought, we observe that line segments PD and QD are of equal length.

Exercise: Show if  D bisects a line segment AB and CD is a line segment with  angles CDA and CDB forming  right angles than C is equidistant from A and B.  

Hint: Apply the SAS isometry criteria. 

The exercise show that each point C on a right bisector of line segment AB is equidistant from line segment endpoints A and B. The line extending  line segment PQ constructed above is a right bisector.