Explore
What is the relationship between the formed quadrilateral EFGH and ABCD, if lines DA. AB, BC, and CD are rotated through the same (blue) angle respectively around vertices A, B, C, and D as shown?
Use the dynamic sketch below to explore this relationship by dragging the indicated points.
1) Click on the 'Show Angles' button. What do you notice?
2) Check your observation in 1) by dragging.
3) Are the two quadrilaterals similar to each other? Or are all EFGH similar to each other?
4) Check your answers in 3) by clicking on the 'Show some Side Ratios' button.
5) Drag either one of A, B, C, D or X. What do you notice about the displayed ratios? What does this tell you?
6) Formulate your observations as a conjecture.
Equi-inclined Lines to the Sides of a Quadrilateral at its Vertices
Conjecture
a) You should've noticed that the angles of EFGH are correspondingly equal to those of ABCD.
b) You should also have noticed that EFGH is not similar to ABCD (ratios of corresponding sides are not constant) and EFGH does not remain similar to itself (ratio of adjacent sides are not constant).
c) But do the corresponding angles of EFGH and ABCD always remain equal?
d) Explore c) further, and observe carefully, when dragging X for a full revolution.
e) What do you notice? Can you explain your observations?
f) Reformulate your conjecture in a) more precisely.
Challenge
7) Can you explain why (prove that) your final conjecture in f) is true? Can you explain (prove) it in more than one way?
8) Can you generalize further to other polygons?
Related Links
A Forgotten Similarity Theorem?
The Equi-inclined Lines to the Angle Bisectors of a Tangential Quadrilateral
The Equi-inclined Bisectors of a Cyclic Quadrilateral
Equi-inclined Lines Problem
Generalizations of a theorem by Wares (click on 'Link to Quadrilateral (equi-inclined lines)')
Further generalizations of Viviani's Theorem (involving Equi-inclined Lines)
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)
Parallelogram Angle Bisectors
The quasi-incentre of a 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
Crossed Quadrilateral Properties
External Links
Similarity (geometry) (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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Created by Michael de Villiers, 19 Nov 2007 with Cinderella; updated to WebSketchpad, 11 June 2025.