Euclidean Geometry

Question

$XYZ$ is a triangle in which $\angle X$ is obtuse. A point $P$ is taken inside the triangle and $XP$, $YP$, $ZP$ are produced to meet the sides $YZ$, $ZX$, $XY$ at the points $K$, $L$, $M$, respectively. Suppose that $PL = PM$.

Find the angles of triangle $XYZ$, given that

• $XK$, $YL$, $ZM$ are the angle bisectors of triangle $XYZ$
• and that $2\;XK = YL$.

Answer 1

hint: make $LQ \perp YZ , Q$ is on $YZ$, find $\dfrac{LQ}{XK}$

Answer 2

The attached drawing show a case where the above requirements are met but I am not sure there is a single solution