,
and 
.
is prime, 
with 
and 
be a function such that for every regular 
-
we have 
.
for all reals 
.
be a triangle 
)
are the altitudes of the triangle. The bisector of 
intersects 
at 
.
be a point such that 
and 
.
passes through the midpoint of 
.
& 
.
Y by Adventure10
Points 
, and 
are on a line in this order. Points 
and 
lie on the same side of this line, in such a way that triangles 
and 
are equilateral. The segments 
and 
intersect in point 
. Prove that 
.
![[asy]
unitsize(1.5 cm);
pair A, B, C, D, E, S;
A = (0,0);
B = (1,0);
C = (2.5,0);
D = dir(60);
E = B + 1.5*dir(60);
S = extension(C,D,A,E);
fill(A--B--D--cycle, gray(0.8));
fill(B--C--E--cycle, gray(0.8));
draw(interp(A,C,-0.1)--interp(A,C,1.1));
draw(A--D--B--E--C);
draw(A--E);
draw(C--D);
draw(anglemark(D,S,A,5));
dot("$A$", A, dir(270));
dot("$B$", B, dir(270));
dot("$C$", C, dir(270));
dot("$D$", D, N);
dot("$E$", E, N);
dot("$S$", S, N);
[/asy]](//latex.artofproblemsolving.com/4/5/5/455249dff7c770632cab86aa44890bd14af670cc.png)
, and 
are on a line in this order. Points 
and 
lie on the same side of this line, in such a way that triangles 
and 
are equilateral. The segments 
and 
intersect in point 
. Prove that 
.![[asy]
unitsize(1.5 cm);
pair A, B, C, D, E, S;
A = (0,0);
B = (1,0);
C = (2.5,0);
D = dir(60);
E = B + 1.5*dir(60);
S = extension(C,D,A,E);
fill(A--B--D--cycle, gray(0.8));
fill(B--C--E--cycle, gray(0.8));
draw(interp(A,C,-0.1)--interp(A,C,1.1));
draw(A--D--B--E--C);
draw(A--E);
draw(C--D);
draw(anglemark(D,S,A,5));
dot("$A$", A, dir(270));
dot("$B$", B, dir(270));
dot("$C$", C, dir(270));
dot("$D$", D, N);
dot("$E$", E, N);
dot("$S$", S, N);
[/asy]](http://latex.artofproblemsolving.com/4/5/5/455249dff7c770632cab86aa44890bd14af670cc.png)
This post has been edited 2 times. Last edited by nsato, Feb 14, 2023, 12:26 AM
