RIDGE

Quadrilateral perpendicular bisector Problem

Description:	Source(s):
Sia dato quadrilatero ABCD. Tracciate gli assi a del lato AB, b del lato BC, c del lato CD, d del lato DA. Sia H il punto di incontro degli assi a e b. K  il punto di incontro degli assi a e d, L  il punto di incontro degli assi c e d, M  il punto di incontro degli assi c e b.  Studiate come varia HKLM al variare di ABCD. Dimostrate le congetture prodotte durante l'esplorazione fatta in Cabri.	Olivero 2002 p. 267
You are given a quadrilateral ABCD. Construct the perpendicular bisectors of its sides: a of AB, b of BC, c of CD, d of DA. H is the intersection of a and b, K of a and d, L of c and d, M of c and b. Investigate how HKLM changes in relation to ABCD. Prove your conjectures.	Olivero 2002 p. 267
A, B, and C are three fixed points. What conditions have to be satisfied by point D for the perpendicular bisectors to the sides of ABCD to meet in a single point?	Marrades R., Gutiérrez Á. (2000)
The perpendicular bisectors of the sides of a quadrilateral Q form a quadrilateral Q1, and the perpendicular bisectors of the sides of Q1 form a quadrilateral Q2. Show that Q2 is similar to Q and find the ratio of similitude.	Bennett, D. (1997), who references Langr (1953)
Under what conditions do the perpendicular bisectors of the sides of a quadrilateral Q form another quadrilateral?	Implicitly in King (1997)

Prior knowledge 

Prior DGS content required:

• perpendicular bisector (tool in Cabri, construction in GSP)

Prior mathematical content required:

• perpendicular bisector

Prior mathematical content related:

• Perpendicular bisectors of triangles.
• Circumcircles

Possible learning

Likely mathematical content learned:

• Cyclic quadrilateral properties

Possible solutions

• They constructed the figure and hid the bisectors
• Taziana  dragged ABCD to make a trapezium and they observed that HKLM also resembled a trapezium.  
• Taziana  dragged ABCD to make a parallelogram and they observed that HKLM also resembled a parallelogram.
• Bartolomeo recorded this conjecture about the parallelogram.   
• Taziana  dragged ABCD to make a rectangle and they observed that HKLM became a point.
• Taziana  dragged ABCD to try to make HKLM bigger. Taziana observed that LM follows what AB does.
• Bartolomeo recorded the conjecture about the rectangle.
• Taziana dragged ABCD to another position in which HKLM became a point.
• Bartolomeo dragged ABCD to make a kite but they did not comment that HKLM also became a kite.
• Bartolomeo dragged ABCD to make a square and observed that the point produced should have been expected because a square is a rectangle.
• Taziana drew a rectangle on paper and exclaimed "of course it's a point"
• Taziana recorded that LM follows what AB does, LK follows what AD does, etc.
• They showed and hid the bisectors.
• Bartolomeo made ABCD a rhombus and recorded the conjecture that if ABCD is a rhombus then HKLM is also a rhombus
• They proved the conjecture for the parallelogram, first by making reference to alternate angles, then simplifying their proof by using the fact that the perpendicular bisectors are perpendiculars to parallel lines. 
• They began to prove the conjecture for the rhombus, then stopped.
• They proved the conjecture for the rectangle, by making reference to the coincidence of the perpendicular bisectors of opposite sides in this case.
• They partly proved the conjecture for the right angled trapezium, by making reference to the perpendicular lines and then using congruent angles to try to establish the similarity of the two trapezia.
• They tried an isosceles trapezium and found the inner quadrilateral was reduced to a point.
• Taziana drew a circle centred on the point and observed that it passed through ABC and D.
• They tried again to prove the conjecture for the rhombus
• They drew a cyclic quadrilateral and observed that the perpendicular bisectors meet in a point. They related this to the property that the opposite angles of a cyclic quadrilateral add up to 180 degrees.

Supported by a research grant from the Social Sciences and Humanities Research Council of Canada

Page last updated July 2008 by David Reid