" ... even if we succeeded in finding a satisfactory solution we may still be interested in finding another solution. We desire to convince ourselves of the validity of a theoretical result by two different derivations as we desire to perceive a material object through two different senses. Having found a proof, we wish to find another proof as we wish to touach an object after having seen it. Two proofs are better than one. 'It is safe riding at two anchors'". - George Polya (1945: 61-62), How to Solve It.
In Round 2 of the South African Mathematics Olympiad 2016, the following problem was posed.
Challenge
1) First try and solve the above problem yourself before continuing below.
Explore More
Interestingly the problem is actually a special case of a theorem related to any right triangle.
2) Click on the 'Show Radii of Circles' button.
3) Drag any of A, B or C and observe how the radii change dynamically. Can you see a relationship between the four radii?
4) Click on the 'Show Sum of Radii' button.
5) What do you notice? Is it always true? Make a conjecture and explore by dragging any of the red vertices.
Generalization of SA Mathematics Olympiad 2016 Problem R2 Q20
Challenge
6) Can you explain why (prove) your observation in 5) is true?
7) Can you prove the result in more than one way?
Published Paper
Read my paper A Multiple Solution Task: Another SA Mathematics Olympiad Problem in Learning and Teaching Mathematics, No. 22, 2017, pp. 42-46 about the problem above.
Further Generalization
It is natural to ask if the same relationship generalizes to any triangle, and if not, if there is another more general relationship relating the four radii to each other?
8) Use your own dynamic software to make constructions and to explore this for yourself.
Check your findings in 8) at this Wolfram MathWorld link: Exradius.
Related Links
SA Mathematics Olympiad Problem 2016, Round 1, Question 20
SA Mathematics Olympiad 2022, Round 2, Q25
IMO 2014 Problem 4 - Geometry
An extension of the IMO 2014 Problem 4
A 1999 British Mathematics Olympiad Problem and its dual
Constant perimeter triangle formed by tangents to circle (SAMO 2013, R2, Q13)
Some Quadrilateral Inequalities involving Sides & Diagonals (1st Th. was used as Q1 in SAMO 2001, R3)
Logical Discovery: Varignon Parallelogram Perimeter (Rethinking Proof activity)
Matric Exam Geometry Problem - 1949 (A variation of Reim's theorem)
Dirk Laurie Tribute Problem
Logical Discovery: Tangential quadrilateral (Pitot's theorem) (Rethinking Proof activity)
Tangential Quadrilateral Converse
Concurrent Angle Bisectors of a Quadrilateral
Extangential Quadrilateral
Triangulated Tangential Hexagon theorem
Theorem of Gusić & Mladinić
Conway's Circle Theorem as special case of Side Divider (Windscreen Wiper) Theorem
Side Divider Theorem for a Tangential Quadrilateral
Bradley's Theorem, its Generalization & an Analogue Theorem
Japanese Circumscribed Quadrilateral Theorem
External Link
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.)
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First created in 2016 with JavaSketchpad by Michael de Villiers, updated to WebSketchpad 28 Sept 2024.