Problem B. 5175. (May 2021)

B. 5175. In a triangle \(\displaystyle ABC\), \(\displaystyle AC=BC\), \(\displaystyle D\) is an interior point of side \(\displaystyle AC\), and \(\displaystyle K\) is the centre of the circle \(\displaystyle ABD\). Show that quadrilateral \(\displaystyle BCDK\) is cyclic.

(3 pont)

Deadline expired on June 10, 2021.

Sorry, the solution is available only in Hungarian. Google translation

Megoldás. \(\displaystyle DAK\) és \(\displaystyle AKB\) egyenlő szárú háromszögek, azaz \(\displaystyle KD=KA=KB\), ezért

\(\displaystyle ADK\sphericalangle = KAD\sphericalangle , \ BAK\sphericalangle = KBA\sphericalangle. \)

Mivel a feltétel szerint \(\displaystyle ACB\) is egyenlőszárú, \(\displaystyle BAC\sphericalangle = CBA\sphericalangle\). Így – előjeles szögekkel számolva:

\(\displaystyle CBK\sphericalangle = CBA\sphericalangle - KBA\sphericalangle = BAC\sphericalangle - BAK\sphericalangle = KAD\sphericalangle = ADK\sphericalangle = 180^{\circ}-KDC \sphericalangle , \)

tehát \(\displaystyle BCDK\) valóban húrnégyszög.

Statistics:

81 students sent a solution.
3 points:	62 students.
2 points:	16 students.
1 point:	2 students.
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Problems in Mathematics of KöMaL, May 2021