Library / Annual Harvard-MIT Mathematics Tournament - Team Round: B Division /1 of 4

Annual Harvard-MIT Mathematics Tournament - Team Round: B Division

Geometry Difficulty 4.6 AIME Prove it United States

Problem:
Let $X$ be the point on side $BC$ such that $BX = CD$ . Show that the excircle $ABC$ opposite of vertex $A$ touches segment $BC$ at $X$ .

Solution

Solution:
Let the excircle touch lines $BC$ , $AC$ and $AB$ at $X'$ , $Y$ and $Z$ , respectively. Using the equal tangent property repeatedly, we have
$BX' - X'C = BZ - CY = (EY - CY) - (FZ - BZ) = CE - BF = CD - BD.$
It follows that $BX' = CD$ , and thus $X' = X$ . So the excircle touches $BC$ at $X$ .

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.