MathDB

8

Part of 2017 All-Russian Olympiad

Problems(2)

Tangents intersect on AC

Source: All Russian 2017,grade 10,day 2,P8

5/1/2017
In a non-isosceles triangle ABCABC,OO and II are circumcenter and incenter,respectively.B′B^\prime is reflection of BB with respect to OIOI and lies inside the angle ABIABI.Prove that the tangents to circumcirle of △BB′I\triangle BB^\prime I at B′B^\prime,II intersect on ACAC. (A. Kuznetsov)
geometrycircumcircleincircleincenterCircumcentergeometric transformationreflection
A nice(Hard) problem from Russia!

Source: All Russian 2017,Grade 11,day 2,P8

5/1/2017
Given a convex quadrilateral ABCDABCD. We denote IA,IB,ICI_A,I_B, I_C and IDI_D centers of ωA,ωB,ωC\omega_A, \omega_B,\omega_C and ωD\omega_D,inscribed In the triangles DAB,ABC,BCDDAB, ABC, BCD and CDACDA, respectively.It turned out that ∠BIAA+∠ICIAID=180∘\angle BI_AA + \angle I_CI_AI_D = 180^\circ. Prove that ∠BIBA+∠ICIBID=180∘\angle BI_BA + \angle I_CI_BI_D = 180^{\circ}. (A. Kuznetsov)
geometryincenterAngle Chasingincircle