Mathematical and Physical Journal
for High Schools
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Problem B. 3809. (March 2005)

B. 3809. The triangle ABC is isosceles, namely AB = BC. The points C1, A1, B1 lie on the sides AB, BC, CA, respectively, and such that \angleBC1A1=\angleCA1B1=\angleCAB. The lines BB1 and CC1 meet at  P. Prove that the quadrilateral AB1PC1 is cyclic.

(4 pont)

Deadline expired on April 15, 2005.


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

Megoldás. Azt kell bizonyítanunk, hogy AC_1C\measuredangle=CB_1B\measuredangle. Mivel C_1AC\measuredangle=B_1CB\measuredangle, ez ekvivalens azzal, hogy az AC1C és CB1B háromszögek hasonlók, ami ugyanezen ok miatt azzal ekvivalens, hogy AC1:AC=CB1:CB, vagyis hogy CB:AC=CB1:AC1. Mivel viszont AC1=A1C, ez azonnal következik az ABC és A1B1C háromszögek hasonlóságából.


Statistics:

89 students sent a solution.
4 points:86 students.
3 points:2 students.
0 point:1 student.

Problems in Mathematics of KöMaL, March 2005