Question 154
Let $ABC$ be a given triangle and $l$ be a line that meets the lines $BC$, $CA$ and $AB$ in $A_1$, $B_1$ and $C_1$ respectively. Let $A'$ be the midpoint of the segment connecting the projections of $A_1$ onto the lines $AB$ and $AC$. Construct analogously the points $B'$ and $C'$.
a. Show that the points $A'$, $B'$ and $C'$ are collinear on some line $l'$.
b. Show that if $l$ contains the circumcenter of the triangle $ABC$, then $l'$ contains the center of its Euler circle.
a. Show that the points $A'$, $B'$ and $C'$ are collinear on some line $l'$.
b. Show that if $l$ contains the circumcenter of the triangle $ABC$, then $l'$ contains the center of its Euler circle.