IMO 2014 Problem 4
The 55th International Mathematical Olympiad (IMO) was held from 3-13 July 2014 in Cape Town. Here is the 1st problem written on the 2nd day of the competition (problem no. 4):
"The points P and Q are chosen on the side BC of a triangle ABC so that ∠PAB = ∠ACB and ∠QAC = ∠CBA. The points M and N are taken on the rays AP and AQ, respectively, so that AP = PM and AQ = QN. Prove that the lines BM and CN intersect on the circumcircle ABC".
Investigate
Drag any of the vertices of ABC.
Challenge
Can you prove the result? Can you prove it in several different ways?
Further Challenge
Also note that the products of the opposite sides of the cyclic quadrilateral ABLC, namely, AB⋅CL and BL⋅AC are equal to half the product of the diagonals AL and BC. Can you also prove this result?
(Note: In the original IMO version, the problem was restricted to an acute-angled triangle to limit the cases for students to consider. Also note that IMO students are not provided with a sketch and have to draw their own.)
IMO 2014 Problem 4 - Geometry
Solutions
Only if stuck, or to compare your proof, click IMO 2014 Problem 4 - Solutions for one possible proof, as well as download links to two PDF files with several more proofs.
Further Generalization
1) Can you generalize the result further to hexagons, etc. by using your observation that the products of the opposite sides of the cyclic quadrilateral ABLC, namely, AB⋅CL and BL⋅ are equal to half the product of the diagonals AL and BC?
2) Think about it before going to: IMO 2014 extension to compare.
Harmonic Quadrilaterals
A cyclic quadrilateral with the products of its opposite sides equal as in the case above is called a harmonic quadrilateral. Read more about these quadrilaterals, their properties & applications to problems in this paper, About the Harmonic Quadrilateral, presented at a conference in Bulgaria in 2012 by Truong, Khanh & Quang from Vietnam.
Harmonic quadrilaterals in turn are special cases of Apollonius quadrilaterals, which are general quadrilaterals with the two products of the lengths of their opposite sides equal.
Some Related Links
An extension of the IMO 2014 Problem 4
Cyclic Quadrilateral Alternate Angles Sum (Rethinking Proof activity)
Cyclic Hexagon Alternate Angles Sum Theorem
A generalization of the Cyclic Quadrilateral Angle Sum theorem
Angle Divider Theorem for a Cyclic Quadrilateral
Side Divider Theorem for a Circumscribed/Tangential Quadrilateral
Cyclic Kepler Quadrilateral Conjectures
The Perpendicular Bisectors of an Apollonius Quadrilateral
Nine-point centre (anticentre or Euler centre) & Maltitudes of Cyclic Quadrilateral
Euler and Nagel lines for Cyclic and Circumscribed Quadrilaterals
Semi-regular Angle-gons and Side-gons: Generalizations of rectangles and rhombi
Alternate sides cyclic-2n-gons and Alternate angles circum-2n-gons
Bradley's Theorem, its Generalization & an Analogue Theorem
Matric Exam Geometry Problem - 1949 (A variation of Reim's theorem)
International Mathematical Talent Search (IMTS) Problem Generalized
SA Mathematics Olympiad Problem 2016, Round 1, Question 20
SA Mathematics Olympiad 2016 Problem R2 Q20
SA Mathematics Olympiad 2022, Round 2, Q25
Some Quadrilateral Inequalities involving Sides & Diagonals (1st Th. was used as Q1 in SAMO 2001, R3)
A 1999 British Mathematics Olympiad Problem and its dual
Similar Parallelograms: A Generalization of a Golden Rectangle property
Ptolemy's theorem
Eight Point Conic for Cyclic Quadrilateral
Geometry Loci Doodling with Cyclic Quadrilaterals
Some External Related Links
Harmonic quadrilateral (Wikipedia)
Apollonius quadrilateral (Wikipedia)
Aimssec Lesson Activities (African Institute for Mathematical Sciences Schools Enrichment Centre)
UCT Mathematics Competition Training Material
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.)
International Mathematical Olympiad (IMO)
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Created by Michael de Villiers, 16 July 2014; modified & updated to WebSketchpad on 4 July 2021; updated 29 Dec 2021; 3 Feb 2023; 1 May 2025; 27 June 2026.