**B. 3832.** *P* is an arbitrary point of the hypotenuse *AB* of a right-angled triangle *ABC*. The foot of the altitude drawn from vertex *C* is *C*_{1}. The projection of *P* onto the leg *AC* is *A*_{1}, and its projection onto the leg *BC* is *B*_{1}.

*a*) Prove that the points *P*, *A*_{1}, *C*, *B*_{1}, *C*_{1} lie on a circle.

*b*) Prove that the triangles *A*_{1}*B*_{1}*C*_{1} and *ABC* are similar.

(3 points)

**Deadline expired on 17 October 2005.**

**Solution. **(a) The angles *PA*_{1}*C*, *PB*_{1}*C*, *PC*_{1}*C* are all right angles, so points *A*_{1}, *B*_{1}, *C*_{1} lie on the circle of diameter *PC*. Due to the right angle at *C*, another diameter of this circle is *A*_{1}*B*_{1}.

(b) From the triangles *ABC* and *CBC*_{1}, *BAC*=90^{o}-*angleABC*=*BCC*_{1}. Since quadrilateral *CA*_{1}*C*_{1}*B*_{1} is cyclic, *B*_{1}*CC*_{1}=*B*_{1}*A*_{1}*C*_{1}. Therefore, the red angles in the Figure are equal.

Similarly, also the blue angles are equal.

Triangles *ABC* and *A*_{1}*B*_{1}*C*_{1} are similar because they have equal angles, respectively.