 Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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# Problem A. 630. (December 2014)

A. 630. The konvex quadrilateral $\displaystyle ABCD$ has an inscribed circle with center $\displaystyle I$. The rays $\displaystyle AB$ and $\displaystyle DC$ meet at point $\displaystyle F$, the rays $\displaystyle AD$ and $\displaystyle BC$ meet at point $\displaystyle G$. Let $\displaystyle \mathcal{E}$ be the ellipse with foci $\displaystyle F$ and $\displaystyle G$ that passes through points $\displaystyle B$ and $\displaystyle D$, and let $\displaystyle \mathcal{H}$ be the hyperbola branch with foci $\displaystyle F$ and $\displaystyle G$ that passes through points $\displaystyle A$ and $\displaystyle C$. Denote by $\displaystyle P$ and $\displaystyle Q$ the intersections of $\displaystyle \mathcal{E}$ and $\displaystyle \mathcal{H}$. Show that the points $\displaystyle P$, $\displaystyle Q$ and $\displaystyle I$ are collinear.

(5 pont)

Deadline expired on January 12, 2015.

### Statistics:

 10 students sent a solution. 5 points: Di Giovanni Márk, Fehér Zsombor, Janzer Barnabás, Lajkó Kálmán, Nagy-György Pál, Papp 893 Marcell, Saranesh Prembabu, Szabó 789 Barnabás, Szőke Tamás, Williams Kada.

Problems in Mathematics of KöMaL, December 2014