Let
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
be any point on the circumscribed circle of
![PQR.](/media/m/1/8/a/18a5d726469c9d62d21e22d57cbfa55d.png)
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
![l(S, PQR).](/media/m/b/0/6/b0697c3534b5d031e1b68fb643065cfb.png)
Suppose that the hexagon
![ABCDEF](/media/m/9/f/e/9fe205b534135e3a700ffb54d8b96cb0.png)
is inscribed in a circle. Show that the four lines
![l(D,ABF),](/media/m/4/9/a/49adb7cbb41d921fd0e0787f86732fdc.png)
and
![l(E,ABC)](/media/m/f/4/f/f4f952e38cd008c276140416fa7a83ad.png)
intersect at one point if and only if
![CDEF](/media/m/e/9/8/e98dd3060893f5715e56a7329cf1adb1.png)
is a rectangle.
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Let $S$ be any point on the circumscribed circle of $PQR.$ 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 $l(S, PQR).$ Suppose that the hexagon $ABCDEF$ is inscribed in a circle. Show that the four lines $l(A,BDF),$ $l(B,ACE),$ $l(D,ABF),$ and $l(E,ABC)$ intersect at one point if and only if $CDEF$ is a rectangle.