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inradius of P A_j A_{j+k} independent of P, if inradii of P A_iA_{i+1} are equal

Source: 15th -b QEDMO problem 10 (19. - 22. 10. 2017) https://artofproblemsolving.com/community/c1512515_qedmo_2005

May 30, 2021
geometryinradiusequal segments

Problem Statement

Let ℓ\ell be a straight line and P∉ℓP \notin \ell be a point in the plane. On ℓ\ell are, in this arrangement, points A1,A2,...A_1, A_2,... such that the radii of the incircles of all triangles PAiAi+1P A_iA_{i + 1} are equal. Let k∈Nk \in N. Show that the radius of the incircle of the triangle PAjAj+kP A_j A_{j + k} does not depend on the choice of j∈Nj \in N .
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