MathDB

2

Part of 2021 European Mathematical Cup

Problems(2)

10th EMC - Angle Trisection

Source: 10th European Mathematical Cup - Problem J2

12/22/2021
Let ABCABC be an acute-angled triangle such that ∣AB∣<∣AC∣|AB|<|AC|. Let XX and YY be points on the minor arc BC{BC} of the circumcircle of ABCABC such that ∣BX∣=∣XY∣=∣YC∣|BX|=|XY|=|YC|. Suppose that there exists a point NN on the segment AY‾\overline{AY} such that ∣AB∣=∣AN∣=∣NC∣|AB|=|AN|=|NC|. Prove that the line NCNC passes through the midpoint of the segment AX‾\overline{AX}. \\ \\ (Ivan Novak)
emcEuropean Mathematical Cuptrisectorgeometrycircumcircle
Nice geometry from EMC

Source: 10th European Mathematical Cup - Problem S2

12/22/2021
Let ABCABC be a triangle and let D,ED, E and FF be the midpoints of sides BC,CABC, CA and ABAB, respectively. Let X≠AX\ne A be the intersection of ADAD with the circumcircle of ABCABC. Let Ω\Omega be the circle through DD and XX, tangent to the circumcircle of ABCABC. Let YY and ZZ be the intersections of the tangent to Ω\Omega at DD with the perpendicular bisectors of segments DEDE and DFDF, respectively. Let PP be the intersection of YEYE and ZFZF and let GG be the centroid of ABCABC. Show that the tangents at BB and CC to the circumcircle of ABCABC and the line PGPG are concurrent.
geometry