Question 5
Let $O$ be the circumcentre of triangle $ABC$, and let $M$ be the midpoint of $AC$. The line $BO$ meets the altitudes $AA_1,CC_1$ at $H_a,H_c$, respectively. The circumcircles of triangles $BH_aA$ and $BH_cC$ meet again at $K$. Prove that $K$ lies on $BM$. (6 points)
Source clarification: the official original specifies an acute triangle; the supplied Vietnamese text omits that adjective. The stated lines, nondegenerate circles and distinct second intersection are understood to exist. The proof below treats the official acute configuration and gives an algebraic extension whenever these objects are defined.
Neutral editorial illustration.
Source clarification: the official original specifies an acute triangle; the supplied Vietnamese text omits that adjective. The stated lines, nondegenerate circles and distinct second intersection are understood to exist. The proof below treats the official acute configuration and gives an algebraic extension whenever these objects are defined.
Neutral editorial illustration.