satisfy that: 
and natural number 
,
Y by Adventure10
Given the acute triangle 
. The circle of diameter 
intersects the altitude 
in 
and 
(with 
), the circle of diameter 
intersects the altitude 
in 
and 
(with 
). Show that the points 
and are concyclic.
. The circle of diameter 
intersects the altitude 
in 
and 
(with 
), the circle of diameter 
intersects the altitude 
in 
and 
(with 
). Show that the points 
and are concyclic.This post has been edited 2 times. Last edited by parmenides51, Oct 24, 2022, 9:39 PM
![[asy]size(250);import olympiad;
pair A=(0,2),B=(-1,0),C=(2,0);
pair D = foot(A,B,C),F = foot(C,A,B),E=foot(B,A,C);
draw(A--B--C--A);draw(A--D,dotted);
pair first[] = intersectionpoints(A..B..A,C--(F+(F-C)*3));
pair second[] = intersectionpoints(A..C..A,B--(E+(E-B)*3));
draw(A..B..A);draw(A..C..A);
pair M=first[1],N=first[0],P=second[0],Q=second[1];
draw(C--N,dotted);draw(B--Q,dotted);
label("$A$",A,dir(90));label("$B$",B,dir(240));label("$C$",C,dir(-60));
label("$D$",D,dir(-90));label("$E$",E,dir(45));label("$F$",F,dir(135));
label("$M$",M,dir(-60));label("$N$",N,dir(135));label("$P$",P,dir(-160));label("$Q$",Q,dir(60));
label("$H$",extension(A,D,B,E),dir(0));
[/asy]](http://latex.artofproblemsolving.com/8/e/a/8ea1a73aeaeb1ad59ef5d52887a57e0985ec2794.png)
to 
be 
.
,
are concyclic,
,
are also concyclic.
all intersect at the orthocenter, 
.
which implies that 
are concyclic.
