Mathematical and Physical Journal
for High Schools
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Problem B. 4085. (April 2008)

B. 4085. Prove that if a symmetrical trapezium has an inscribed circle then its height is the geometric mean of the bases.

(3 pont)

Deadline expired on May 15, 2008.

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

Megoldás: Jelölje az alapok hosszát a,b, a szárakét c, a magasságét pedig m. Mivel a trapéz érintőnégyszög, a+b=2c. A Pithagorasz-tétel alapján ezért

m^2=c^2-\Bigl(\frac{a-b}{2}\Bigr)^2=\Bigl(\frac{a+b}{2}\Bigr)^2 -\Bigl(\frac{a-b}{2}\Bigr)^2=ab,

amint azt igazolnunk kellett.

Statistics:

149 students sent a solution.
3 points:139 students.
2 points:9 students.
1 point:1 student.

Problems in Mathematics of KöMaL, April 2008