The construction of the angle bisectors of certain non-tangential quadrilaterals produces the following interesting known results:
1) for a parallelogram, a rectangle is formed
2) for a rectangle, a square is formed
3) for a general quadrilateral, a cyclic quadrilateral is formed
4) regarding the cyclic quadrilateral in 3): the intersection of its diagonals (called the quasi-incentre) is equi-distant from each pair of opposite sides of the original quadrilateral.
Note
Dynamic geometry activities for the first three results mentioned above are available in my book Rethinking Proof (free to download) as well as online at this link: Parallelogram Angle Bisectors. For an online, dynamic activity for the fourth result, go to: The quasi-incentre of a quadrilateral.
What if?
In all of the above it is assumed that the quadrilateral is non-tangential, since the angle bisectors of a tangential (circumscribed) quadrilateral would be concurrent (at the incentre), and no quadrilateral is formed.
But what if instead of angle bisectors of a tangential quadrilateral, we construct 'equi-inclined lines' through the vertices, i. e. lines that make equal angles with the angle bisectors? What sort of quadrilateral is formed?
Explore
The dynamic sketch below shows a tangential quadrilateral ABCD with equi-inclined lines to the angle bisectors and though the vertices 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 Measurements' button. What do you notice?
3) What do you notice when you drag either one of the tangential points I, J, K or L?
4) What do you notice when you keep the shape of ABCD fixed, and just drag point X to change the angle of the equi-inclined lines?
5) Do the results also hold if ABCD concave?
6) Formulate your observed conjectures.
Equi-inclined Lines to Angle Bisectors of a Tangential Quadrilateral
Conjecture
You should've noticed the following:
a) the equi-inclined lines to the angle bisectors of tangential ABCD form a cyclic quadrilateral A'B'C'D'
b) if ABCD remains fixed, then all cyclic quadrilaterals A'B'C'D' are similar to each other (note that its angles and the ratio of its sides remain constant as displayed.
Challenge
7) Can you prove these results? Can you prove them in more than one way?
(Hint: Click on the 'Link to Lemma' button which shows a quadrilateral formed by perpendiculars to the inradii at the vertices of the tangential quadrilateral ABCD. You may find it useful in explaining/proving your observations).
Further Generalization to Tangential Polygons
8) Can you generalize further to a tangential pentagon?
9) Check your answer to 8) by clicking on the 'Link to tangential pentagon' button to navigate to a new sketch showing a tangential pentagon.
10) Click on the 'Show Perpendicular Bisectors' button. What do you notice? What can you conclude from that?
11) Click on the 'Show Angles & Side Ratios' button.
12) What do you notice about the angles & side ratios of A'B'C'D'E' when you keep the shape of ABCDE fixed, and just drag point X to change the angle of the equi-inclined lines?
13) Can you explain/prove your observations in 12)?
14) Can you generalize further to tangential hexagons, etc.?
Further Generalization/Variation to General Polygons
As mentioned at the top of the page: it is well known that the angle bisectors of a (non-tangential) quadrilateral form a cyclic quadrilateral. But what happens if we draw similar equi-inclined lines to the angle bisectors of an arbitrary quadrilateral? What shape is formed by the equi-inclined lines in this case?
15) Click on the 'Link to general quad' button.
(In this sketch, the angle bisectors are shown by blue dashed lines and the equi-inclined lines by pink dashed lines).
16) What do you notice about the formed quadrilateral A'B'C'D'?
17) Click on the 'Show Angles & Side Ratios' button.
18) What do you notice about the angles & side ratios of A'B'C'D' when you keep the shape of ABCD fixed, and just drag point X to change the angle of the equi-inclined lines?
19) Can you explain/prove your observations in 16) & 18)?
Explore More: Do the observed properties in 16) and 18) generalize to an arbitrary pentagon?
20) Click on the 'Link to general pentagon' button.
21) Click on the 'Show Circles' button which show circles drawn through three vertices of the pentagons formed respectively by the angle bisectors and the equi-inclined lines. What do you notice?
22) Click on the 'Show Angles & Side Ratios' button.
23) What do you notice about the angles & side ratios of A'B'C'D'E' when you keep the shape of ABCDE fixed, and just drag point X to change the angle of the equi-inclined lines?
24) Can you explain/prove your observations in 23)?
25) Can you generalize further to arbitrary hexagons, etc.?
Analogous Result
For an analogous result to this tangential quadrilateral result, go to: The Equi-inclined Bisectors of a Cyclic Quadrilateral.
(Here the concept of 'lines through the vertices, equi-inclined to the angle bisectors of a tangential quadrilateral' is replaced by the analogous one of 'equi-inclined bisectors of a cyclic quadrilateral').
Submitted Paper
A paper on these results by Hans Humenberger, University of Vienna, and myself has been submitted for publication. All rights reserved.
Related Links
A Forgotten Similarity Theorem?
Parallelogram Angle Bisectors (Rethinking Proof Activity)
The quasi-incentre of a quadrilateral
Equi-inclined Lines to the Sides of a Quadrilateral at its Vertices
The Equi-inclined Bisectors of a Cyclic Quadrilateral
Concurrent Angle Bisectors of a Quadrilateral
Cyclic Quadrilateral Angle Bisectors Rectangle Result
Cyclic Quadrilateral Incentres Rectangle (Japanese theorem for cyclic quadrilaterals)
Angle Divider Theorem for a Cyclic Quadrilateral
Pitot's Theorem for a tangential quadrilateral
Equi-inclined Lines Problem
A variation of Miquel's theorem and its generalization
A generalization of Neuberg's Theorem and the Simson line
Crossed Quadrilateral Properties
External Links
Tangential quadrilateral (Wikipedia)
Cyclic 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, 8 June 2025; updated 9 June 2025; 18 June 2025; 8 August 2025.