IMO Shortlist 1991 problem 3
Kvaliteta:
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Avg: 0,0 Let
be any point on the circumscribed circle of
Then the feet of the perpendiculars from S to the three sides of the triangle lie on the same straight line. Denote this line by
Suppose that the hexagon
is inscribed in a circle. Show that the four lines
and
intersect at one point if and only if
is a rectangle.
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
![PQR.](/media/m/1/8/a/18a5d726469c9d62d21e22d57cbfa55d.png)
![l(S, PQR).](/media/m/b/0/6/b0697c3534b5d031e1b68fb643065cfb.png)
![ABCDEF](/media/m/9/f/e/9fe205b534135e3a700ffb54d8b96cb0.png)
![l(A,BDF),](/media/m/b/a/3/ba3b8f0976ef6d92825b9f340fcc3f27.png)
![l(B,ACE),](/media/m/9/3/5/935e408b39b7db991cd3b6491b604c9b.png)
![l(D,ABF),](/media/m/4/9/a/49adb7cbb41d921fd0e0787f86732fdc.png)
![l(E,ABC)](/media/m/f/4/f/f4f952e38cd008c276140416fa7a83ad.png)
![CDEF](/media/m/e/9/8/e98dd3060893f5715e56a7329cf1adb1.png)
Izvor: Međunarodna matematička olimpijada, shortlist 1991