Concurrence of lines defined by intersections of circles
by Lukaluce, Apr 14, 2025, 10:57 AM
Let 
be an acute-angled triangle and 
, and 
be the feet of the altitudes from 
, and 
, respectively. On the rays 
, and 
, we have points 
, and 
respectively, lying outside of , such that
![\[\frac{A_1A_2}{AA_1} = \frac{B_1B_2}{BB_1} = \frac{C_1C_2}{CC_1}.\]](//latex.artofproblemsolving.com/d/f/a/dfa94e10638971d53066c20c4589a9c125059911.png)
If the intersections of 
and 
, 
and 
, and 
and 
are 
, and 
respectively, prove that 
, and 
have a common point.
be an acute-angled triangle and 
, and 
be the feet of the altitudes from 
, and 
, respectively. On the rays 
, and 
, we have points 
, and 
respectively, lying outside of , such that![\[\frac{A_1A_2}{AA_1} = \frac{B_1B_2}{BB_1} = \frac{C_1C_2}{CC_1}.\]](http://latex.artofproblemsolving.com/d/f/a/dfa94e10638971d53066c20c4589a9c125059911.png)
If the intersections of 
and 
, 
and 
, and 
and 
are 
, and 
respectively, prove that 
, and 
have a common point.
light bulbs are arranged around a circle, and are consecutively numbered with
.
days we change the current on/off configuration as
,
-
natural numbers 
,
be a triangle with 
and the rea is 
Find the minimum of 
be the feet of the altitudes of the triangle 
,
is the point on the circumcircle of the triangle 
is the orthocenter of the triangle 
has radius 
.
be the point such that 
,
is perpendicular to the line 
and 
similarly. Prove that 
are collinear.
,
be all the positive integers smaller than 
such that
for all 
.
and the circle 
of radius 
centered at 
Let 
be the midpoint of 
The line 
intersects 
Let 
be a point on 
Let 
and 
such that 
is a perfect square.
,
,
.
.
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