is the incentre and point 
is defined such that 
and 
is the perpendicular bisector of 
with 
.
as 
,
and 
as 
.
.
Y by Adventure10
In the right-angled triangle 
, the bisectors 
and 
are drawn from the vertices of acute angles. The points 
and 
are the feet of the perpendiculars drawn from 
and 
on the hypotenuse 
. Find the angle 
.
, the bisectors 
and 
are drawn from the vertices of acute angles. The points 
and 
are the feet of the perpendiculars drawn from 
and 
on the hypotenuse 
. Find the angle 
.This post has been edited 3 times. Last edited by parmenides51, Dec 19, 2022, 11:58 AM
![[asy]
size(8cm);
defaultpen(fontsize(10pt));
pair A = (0,3), B = (4,0), C = origin, I = incenter(A,B,C), P = extension(A,I,B,C), Q = extension(B,I,A,C), D = foot(Q,A,B), E = foot(P,A,B);
dot("$A$", A, dir(135));
dot("$B$", B, dir(315));
dot("$C$", C, dir(225));
dot("$D$", D, dir(45));
dot("$E$", E, dir(45));
dot("$P$", P, dir(270));
dot("$Q$", Q, dir(180));
draw(A--B--C--cycle);
draw(A--P--E);
draw(B--Q--D);
draw(D--C--E, dotted);
draw(circumcircle(A,C,E), dotted);
draw(circumcircle(B,C,D), dotted);
[/asy]](http://latex.artofproblemsolving.com/0/a/5/0a58602ea287e51aa50cbac7e30de27f42ca9565.png)
