SEAMO 2027 - Paper E - SET 10 - Self-study Question 18

In triangle $ABC$, points $B^{\prime},C^{\prime}$ lie on $AB,AC$ with $BB^{\prime}=CC^{\prime}$. Let $O,O^{\prime}$ be the circumcentres of $ABC$ and $AB^{\prime}C^{\prime}$. The line $OO^{\prime}$ meets the lines $AB^{\prime},AC^{\prime}$ at $B^{\prime\prime},C^{\prime\prime}$, respectively. Given $AB=\tfrac12AC$, which statement is true?

Use the nondegenerate construction shown: the two circumcentres are distinct and the indicated line intersections exist. Lines may be extended.