Let
be a fixed triangle, and let
,
,
be the midpoints of sides
,
,
, respectively. Let
be a variable point on the circumcircle. Let lines
,
,
meet the circumcircle again at
,
,
, respectively. Assume that the points
,
,
,
,
,
are distinct, and lines
,
,
form a triangle. Prove that the area of this triangle does not depend on
.
Author: Christopher Bradley, United Kingdom
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Let $ABC$ be a fixed triangle, and let $A_1$, $B_1$, $C_1$ be the midpoints of sides $BC$, $CA$, $AB$, respectively. Let $P$ be a variable point on the circumcircle. Let lines $PA_1$, $PB_1$, $PC_1$ meet the circumcircle again at $A'$, $B'$, $C'$, respectively. Assume that the points $A$, $B$, $C$, $A'$, $B'$, $C'$ are distinct, and lines $AA'$, $BB'$, $CC'$ form a triangle. Prove that the area of this triangle does not depend on $P$.
Author: Christopher Bradley, United Kingdom