MathDB

#4

Part of Mexican Quarantine Mathematical Olympiad

Problems(1)

Orthocenter is the midpoint of the altitude

Source: Mexican Quarantine Mathematical Olympiad P4

4/26/2020
Let ABCABC be an acute triangle with orthocenter HH. Let A1A_1, B1B_1 and C1C_1 be the feet of the altitudes of triangle ABCABC opposite to vertices AA, BB, and CC respectively. Let B2B_2 and C2C_2 be the midpoints of BB1BB_1 and CC1CC_1, respectively. Let OO be the intersection of lines BC2BC_2 and CB2CB_2. Prove that OO is the circumcenter of triangle ABCABC if and only if HH is the midpoint of AA1AA_1.
Proposed by Dorlir Ahmeti
geometryaltitudeorthocenterCircumcenter