AoPS - Problem

Source: Romanian Junior BkMO TST 2004, problem 3, created by Fianu M

Tags: geometry, geometry proposed

Valentin Vornicu

01.05.2004 20:05

Let $V$ be a point in the exterior of a circle of center $O$, and let $T_1,T_2$ be the points where the tangents from $V$ touch the circle. Let $T$ be an arbitrary point on the small arc $T_1T_2$. The tangent in $T$ at the circle intersects the line $VT_1$ in $A$, and the lines $TT_1$ and $VT_2$ intersect in $B$. We denote by $M$ the intersection of the lines $TT_1$ and $AT_2$. Prove that the lines $OM$ and $AB$ are perpendicular.

$A$ is the pole of $TT_1$, so $OA\perp TT_1=BM$. $A$ is the pole of $TT_1$ and $T_2$ is the pole of $VT_2$, so $AT_2$ is the polar of $B=TT_1\cap VT_2$, so $OB\perp AT_2=AM$. From here we find $M$ to be the orthocenter of $OAB$ and we're done. I know it's not the solution you were talking about, because (a) it's >1 line long and (b) it uses poles, so it's not suitable for 7'th or 8'th graders , but it's a solution, right? Anyway, it's obvious that $OA\perp TT_1$, so all we need to do is to find a short argument for the fact that $OB\perp AT_2$. I didn't think about this after seeing that it works out fine with poles ..