Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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Problem A. 613. (March 2014)

A. 613. There is given a convex quadrilateral $\displaystyle ABCD$. The points $\displaystyle E$ and $\displaystyle F$ lie on the line segment $\displaystyle AB$, $\displaystyle G$ and $\displaystyle H$ lie on $\displaystyle BC$, $\displaystyle I$ and $\displaystyle J$ lie on $\displaystyle CD$, and $\displaystyle K$ and $\displaystyle L$ lie on $\displaystyle DA$ in such a way that $\displaystyle AE<AF<AB$, $\displaystyle BG<BH<BC$, $\displaystyle CI<CJ<CD$, and $\displaystyle DK<DL<DA$. The line $\displaystyle EJ$ meets $\displaystyle GL$ and $\displaystyle HK$ at $\displaystyle P$ and $\displaystyle S$, and $\displaystyle FI$ meets $\displaystyle GL$ and $\displaystyle HK$ at $\displaystyle Q$ and $\displaystyle R$, respectively. The points $\displaystyle P$ and $\displaystyle R$ lie on the diagonal $\displaystyle AC$ and the points $\displaystyle Q$ and $\displaystyle S$ lie on $\displaystyle BD$ (see the first cover). Suppose that each of the quadrilaterals $\displaystyle AEPL$, $\displaystyle BGQF$ and $\displaystyle CIRH$ has an inscribed circle.

Show that the quadrilateral $\displaystyle DKSJ$ also has an inscribed circle.

Based on a problem of the International Zhautykov Olympiad

(5 pont)

Deadline expired on April 10, 2014.

Statistics:

 1 student sent a solution. 4 points: Fehér Zsombor.

Problems in Mathematics of KöMaL, March 2014