MathDB

Problem 3

Part of 1994 Korea National Olympiad

Problems(2)

identities with triangle of excenters

Source: Korean Mathematical Olympiad 1994, Final Round P3 FKMO

7/21/2018
In a triangle ABCABC, II and OO are the incenter and circumcenter respectively, A′,B′,C′A',B',C' the excenters, and O′O' the circumcenter of △A′B′C′\triangle A'B'C'. If RR and R′R' are the circumradii of triangles ABCABC and A′B′C′A'B'C', respectively, prove that: (i) R′=2RR'= 2R (ii) IO′=2IOIO' = 2IO
geometryexcentersincentercircumradius
cos\alpha : cos\beta : cos\gamma = 39 : 33 : 25

Source: Korean Mathematical Olympiad 1994, Final Round P6 FKMO

7/21/2018
Let α,β,γ\alpha,\beta ,\gamma be the angles of △ABC\triangle ABC. a) Show that cos2α+cos2β+cos2γ=1−2cosαcosβcosγcos^2\alpha +cos^2\beta +cos^2 \gamma =1-2cos\alpha cos\beta cos\gamma . b) Given that cosα:cosβ:cosγ=39:33:25cos\alpha : cos\beta : cos\gamma = 39 : 33 : 25, find sinα:sinβ:sinγsin\alpha : sin\beta : sin\gamma .
Trigonometric Identitiestrigonometrygeometry