MathDB

6

Part of 2011 Regional Olympiad of Mexico Center Zone

Problems(1)

sum of inradii of circles inscribed in curvilinear triangle <= sum of radius

Source: Mathematics Regional Olympiad of Mexico Center Zone 2011 P6

11/11/2021
Given a circle CC and a diameter ABAB in it, mark a point PP on ABAB different from the ends. In one of the two arcs determined by ABAB choose the points MM and NN such that ∠APM=60∘=∠BPN\angle APM = 60 ^ \circ = \angle BPN. The segments MPMP and NPNP are drawn to obtain three curvilinear triangles; APMAPM , MPNMPN and NPBNPB (the sides of the curvilinear triangle APMAPM are the segments APAP and PMPM and the arc AMAM). In each curvilinear triangle a circle is inscribed, that is, a circle is built tangent to the three sides. Show that the sum of the radii of the three inscribed circles is less than or equal to the radius of CC.
geometryinradiusgeometric inequality