MathDB

3

Part of 2024 Francophone Mathematical Olympiad

Problems(2)

Geometry with many parallels and perpendiculars

Source: Francophone 2024, Junior P3

4/4/2024
Let ABCABC be an acute triangle with AB<ACAB<AC and let OO be its circumcenter. Let DD be a point on the segment ACAC such that AB=ADAB=AD. Let EE be the intersection of the line ABAB with the perpendicular line to AOAO through DD. Let FF be the intersection of the perpendicular line to OCOC through CC with the line parallel to ACAC and passing through EE. Finally, let the lines CECE and DFDF intersect in GG. Show that AGAG and BFBF are parallel.
geometrygeometry proposed
Parallel lines in two-circle configuration

Source: Francophone 2024, Senior P3

4/4/2024
Let ABCABC be an acute triangle, ω\omega its circumcircle and OO its circumcenter. The altitude from AA intersects ω\omega in a point D≠AD \ne A and the segment ACAC intersects the circumcircle of OCDOCD in a point E≠CE \ne C. Finally, let MM be the midpoint of BEBE. Show that DEDE is parallel to OMOM.
geometrygeometry proposedcircumcircle