## A 1999 British Mathematics Olympiad Problem and its dual

Theorem 1: If the perpendicular bisectors of the alternate sides AB, CD & EF of a (convex) hexagon ABCDEF circumscribed around a circle are concurrent at the centre of the circle, then the lines PY, XR & QZ connecting opposite tangential points are concurrent.

Theorem 2 (dual): If the angle bisectors of the alternate angles B, D & F of a (convex) hexagon ABCDEF inscribed in a circle are concurrent at the centre of the circle, then the lines AD, BE & CF connecting opposite vertices are concurrent.

Corollary 1: If the perpendicular bisectors of the alternate sides AB, CD & EF of a (convex) hexagon ABCDEF circumscribed around a circle are concurrent at the centre of the circle, then it has 3 pairs of equal adjacent angles (e.g. angle A = angle B, angle C = angle D , and angle E = angle F).

Corollary 2: If the angle bisectors of the alternate angles B, D & F of a (convex) hexagon ABCDEF inscribed in a circle are concurrent at the centre of the circle, then it has 3 pairs of equal adjacent sides (e.g. EF = FA, AB = BC, and CD = DF).

#### .sketch_canvas { border: medium solid lightgray; display: inline-block; } A 1999 British Mathematics Olympiad Problem and its dual

Challenge: Can you explain why (prove) the results are true? If stuck, have a look at my 2002 Math Gazette paper at A dual to a BMO problem.

Note: Theorem 2 follows directly from Corollary 2 with the application of this lovely theorem by Cartensen (2000-2001): The main diagonals of a cyclic hexagon are concurrent, if and only if, the two products of alternate sides are equal; i.e. NB⋅LC⋅MA = BL⋅CM⋅AN. This theorem also appears in Gardiner & Bradley (2002, p. 96; 99) and also on the Math Stack Exchange (2013).
References
Cartensen, J. (2000-2001). About hexagons. Mathematical Spectrum, 33(2), pp. 37–40.
Gardiner, A.D. & Bradley, C.J. (2005). Plane Euclidean Geometry: Theory and Problems. Leeds, University of Leeds: The United Kingdom Mathematics Trust.
Math Stack Exchange. (2013). Accessed on 31 July 2021 at: diagonals of cyclic hexagon.

Further Result: Clicking on the 'Show Diagonals' button, reveals another interesting corollary of the 1st result, namely, that the main diagonals are concurrent at the same point of concurrency as the lines PY, XR & QZ. In general, according to Brianchon's theorem the diagonals of any hexagon circumscribed around a conic are concurrent. So in this case, we have that the lines PY, XR & QZ are concurrent at the Brianchon point.

Further Result Challenge: Can you explain why (prove) the BMO-Brianchon result above is also true? If stuck, have a look at the neat 2005 proof of my friend & colleague, Michael Fox, from the United Kingdom at BMO-Brianchon Concurrency problem. An accompanying dynamic sketch that illustrates the proof is at dynamic proof illustration.

Further Generalizations
1) The first result can be generalized to a hexagon circumscribed around a conic with the midpoints of the alternate sides AB, CD & EF of a (convex) hexagon ABCDEF touching the conic. For a dynamic sketch illustrating this generalization go to BMO Conic General problem.
2) Corollary 2 of the second result can be generalized to a cyclic hexagon with concurrent diagonals, which then has its two products of alternate sides equal, as shown at BMO 'Dual' General problem.

Created by Michael de Villiers, 30 October 2010; updated to WebSketchpad 6 May 2021.