MathDB

8

Part of 2016 All-Russian Olympiad

Problems(2)

Circumcircles are tangent

Source: All russian olympiad 2016,Day2,grade 10,P8

5/1/2016
In acute triangle ABCABC,AC<BCAC<BC,MM is midpoint of ABAB and Ω\Omega is it's circumcircle.Let C′C^\prime be antipode of CC in Ω\Omega. AC′AC^\prime and BC′BC^\prime intersect with CMCM at K,LK,L,respectively.The perpendicular drawn from KK to AC′AC^\prime and perpendicular drawn from LL to BC′BC^\prime intersect with ABAB and each other and form a triangle Δ\Delta.Prove that circumcircles of Δ\Delta and Ω\Omega are tangent.(M.Kungozhin)
geometry
Circumcircles intersect at one point

Source: All russian olympiad 2016,Day2,grade 11,P8

5/1/2016
Medians AMA,BMB,CMCAM_A,BM_B,CM_C of triangle ABCABC intersect at MM.Let ΩA\Omega_A be circumcircle of triangle passes through midpoint of AMAM and tangent to BCBC at MAM_A.Define ΩB\Omega_B and ΩC\Omega_C analogusly.Prove that ΩA,ΩB\Omega_A,\Omega_B and ΩC\Omega_C intersect at one point.(A.Yakubov) [hide=P.S]sorry for my mistake in translation :blush: :whistling: .thank you jred for your help :coolspeak:
geometrycircumcirclegeometry proposedmedian