Mathematical and Physical Journal
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Problem C. 991. (May 2009)

C. 991. A right-angled triangle of sides 3, 4, 5 is cut into two parts with a line perpendicular to the hypotenuse. One part is a quadrilateral that has an inscribed circle, and the other part is a right-angled triangle. Find the lengths of the sides of the quadrilateral.

(5 pont)

Deadline expired on June 15, 2009.


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

Megoldás. Legyen a=3, b=4, c=5.

1.eset: Az egyenes a b oldalt metszi a Qb pontban.

Legyen BP=x, ekkor PA=5-x. Mivel AQ_bP\triangle\approx ABC\triangle, ezért \frac{PQ_b}{5-x}=\frac34~\Rightarrow~PQ_b=\frac34(5-x) és \frac{Q_bA}{5-x}=\frac54~\Rightarrow~Q_bA=\frac54(5-x).

Ekkor CQ_b=4-\frac54(5-x)=\frac{5x-9}{4}.

Mivel BCQbP érintőnégyszög, azért BC+QbP=CQb+PB, azaz 3+\frac34(5-x)=\frac{5x-9}{4}+x, ebből x=3.

Tehát a négyszög oldalai 3; 1,5; 1,5; 3.

2. eset: Az egyenes az a oldalt metszi a Qa pontban. Legyen AP=x, ekkor PB=5-x. Mivel BQ_aP\triangle\approx BAC\triangle, ezért \frac{PQ_a}{5-x}=\frac43~\Rightarrow~PQ_a=\frac43(5-x) és \frac{Q_aB}{5-x}=\frac53~\Rightarrow~Q_aB=\frac53(5-x).

Ekkor CQ_a=3-\frac53(5-x)=\frac{5x-16}{3}.

Mivel APQaC érintőnégyszög, azért AC+QaP=CQa+PA, 4+\frac43(5-x)=\frac{5x-16}{3}+x, ebből x=4.

Tehát a négyszög oldalai 4; \frac43; \frac43; 4.


Statistics:

123 students sent a solution.
5 points:76 students.
4 points:5 students.
3 points:30 students.
2 points:8 students.
1 point:3 students.
Unfair, not evaluated:1 solution.

Problems in Mathematics of KöMaL, May 2009