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 $2\;XK = YL$.

I proved that XYZ is an isosceles triagle; XZ=XY. That means the angle bisector of X will be Perpendicular to YZ (and median). — Lama, Jan 06 '16 at 20:43
After this I get confused. . I dont know how to make pythagoras law and sine law usuful in sovling my problem! — Lama, Jan 06 '16 at 20:54
I placed a drawing in the answer section that meets the above requirements without being isosceles triangle. Could you provide the proof for your claim? — Moti, Jan 09 '16 at 02:56

score 0 · Answer 1 · answered Jan 07 '16 at 01:29

0

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

answered Jan 07 '16 at 01:29

chenbai

7,581

score 0 · Answer 2 · answered Jan 09 '16 at 03:02

0

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

answered Jan 09 '16 at 03:02

Moti

1,928

Euclidean Geometry

2 Answers2