The construction of the perpendicular bisectors of certain non-cyclic quadrilaterals produces the following interesting known results:
1) for a trapezoid, a trapezoid similar to the original is formed - see Some Trapezoid (Trapezium) Explorations (See Investigation 5)
2) for a tangential quadrilateral, another tangential quadrilateral is formed - see Perpendicular Bisectors of Tangential Quadrilateral
3) for an Apollonius quadrilateral, another Apollonius quadrilateral is formed - see The Perpendicular Bisectors of an Apollonius Quadrilateral
4) for a general quadrilateral, another quadrilateral is formed where the intersection of its diagonals (called the quasi-circumcentre) is equi-distant from each pair of opposite vertices of the original quadrilateral - see The quasi-circumcentre of a quadrilateral.
What if?
In all of the above it is assumed that the quadrilateral is non-cyclic, since the perpendicular bisectors of a cyclic quadrilateral would be concurrent (at the circumcentre), and no quadrilateral is formed.
But what if instead of perpendicular bisectors to the sides of a cyclic quadrilateral, we construct 'equi-inclined lines' through the midpoints, i. e. lines that make equal angles with the perpendicular bisectors (or equivalently, make equal angles with the sides)? What sort of quadrilateral is formed?
Explore
The dynamic sketch below shows a cyclic quadrilateral ABCD with equi-inclined lines though the midpoints of its sides forming another quadrilateral A'B'C'D'.
1) Drag the indicated points in the sketch to explore the properties of the formed quadrilateral. What do you notice?
2) Click on the 'Show Objects' button. What do you notice?
3) Does the result also hold if ABCD crossed?
4) Formulate a conjecture.
Equi-inclined Bisectors of a Cyclic Quadrilateral
Conjecture
You should've noticed that the equi-inclined bisectors of ABCD form another cyclic quadrilateral A'B'C'D' similar to the original, and that its circumcircle is concentric with that of the original.
Challenge
5) Can you prove the result? Can you prove it in more than one way?
Further Generalization
6) Can you generalize further?
7) Check your answer to 6) by clicking on the 'Link to cyclic pentagon' button to navigate to a new sketch showing a cyclic pentagon.
8) What do you notice? Can you explain/prove your results?
9) Can you generalize further?
Even Further Generalization
What happens if we do the same equi-inclined bisector construction on an arbitrary quadrilateral?
10) Click on the 'Link to general quad' button to navigate to a new sketch showing a general quadrilateral with equi-inclined bisectors of its sides.
11) What do you notice? Can you explain/prove your results?
12) Can you generalize further?
Analogous Result
For an analogous result to this cyclic quadrilateral result, go to: The Equi-inclined Lines to the Angle Bisectors of a Tangential Quadrilateral.
(Here the concept of 'equi-inclined bisectors of a cyclic quadrilateral' is replaced by the analogous one of 'lines through the vertices, equi-inclined to the angle bisectors of a tangential quadrilateral').
Submitted Paper
A paper on this result by Hans Humenberger, University of Vienna, and myself has been submitted for publication. All rights reserved.
Related Links
Water Supply: Four Towns (introduction to perpendicular bisectors)
Water Supply: Three Towns (concurrency of perpendicular bisectors)
A variation of Miquel's theorem and its generalization (uses equi-inclined lines)
A generalization of Neuberg's Theorem & the Simson line (using equi-inclined lines)
Equi-inclined Lines Problem
Generalizations of a theorem by Wares (click on 'Link to Quadrilateral (equi-inclined lines)')
Distances in an Equilateral Triangle (Viviani's theorem)
2D Generalizations of Viviani's Theorem
Further generalizations of Viviani's Theorem (involving Equi-inclined Lines)
A Forgotten Similarity Theorem?
The quasi-circumcentre of a quadrilateral
A theorem involving the perpendicular bisectors of a hexagon with opposite sides parallel
Perpendicular Bisectors of Circumscribed/Tangential Quadrilateral Theorem
The Equi-inclined Lines to the Angle Bisectors of a Tangential Quadrilateral
Equi-inclined Lines to the Sides of a Quadrilateral at its Vertices
Conway’s Circle Theorem as special case of Windscreen Wiper Theorem
A 1999 British Mathematics Olympiad Problem and its dual
Some Trapezoid (Trapezium) Explorations (See Investigation 5)
Carnot's (or Bottema's) Perpendicularity Theorem & Some Generalizations
External Links
Cyclic quadrilateral (Wikipedia)
Tangential quadrilateral (Wikipedia)
Spiral similarity (Wikipedia)
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)
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Michael de Villiers, created with WebSketchpad, 1 June 2025; updated 5/9 June 2025; 19 June 2025.