D2906 – Le pentagone cyclophile Solution de V. Prasolov (extraite de l’ouvrage « Problems in plane geometry ») Première étape Démonstration

D2906 – Le pentagone cyclophile

Solution de V. Prasolov (extraite de l’ouvrage « Problems in plane geometry »)

Première étape

Démonstration

Deuxième étape

The sides of convex pentagon ABCDE are extended so that five-angled star AHBKCLDMEN is formed. The circles are circumscribed about triangles – the rays of the star.

Prove that the five intersection points of these circles distinct from A,B,C,D,E lie on one circle.

Démonstration

Troisième étape

Let P,Q,R,S,T the intersection points of circles S₁,S₂,S₃,S₄,S₅ spoken about in the formulation of the problem.

Let us prove, for instance, that points P,Q,R,S lie on one circle. Let us draw circle Σ circumscribed about triangle NKD. Applying the resulat of Problem 2.83 a) (see appendix) to quadrilaterals AKDE and BNDC we see that circles S₄, S₅ and Σ intersect at one point (namely, P) and circles S₂,S₃ and Σ also intersect at one point (namely, S)

Therefore, circle Σ passes through points P and S. Now observe that of eight intersection points of circles Σ,S₁,S₂,S₅, four, namely, N, A, B, K, lie on one line. If follow that by problem 28.31, the remaining four points P,Q,R,S lie on one circle.

Appendix

Démonstration