Question

In a triangle PQR x and y are the points on PQ and QR respectively.If PQ=QR and QX=QY,show that PX=RY

Answer 1

Let P be the perimetre.
P=PQ+QR+RP
P(PX+XQ)+(QY+YR)+(RZ+ZP)
When the circle touches the triangle at X,Y &Z then it obviously means that PQ,QR,&PR are tangents to the circle.
Hence,let 'O' be the center of the circle.
OX,OY &OZ  are the radii.The angle between a tangent and radius is 90°.
OXZ is isosceles triangle due to OX=OZ
∠OZX=∠OXZ
∠PXO=∠PZO=9O°
∠PXZ+∠OXZ=∠PZX+∠OZX
But ∠OZX=∠OXZ
Therefore,∠PXZ=∠PZX
Triangle PXZ is isosceles and so PX=PZ
Therefore,QX=QY&YR=RZ
P=2PX+2QY+2RZ(Substitute XQ=QY,YR=RZ,&ZP=PX)
P=2(PX+QY+RZ) or PX+QY+RZ=1/2P
QY=XQ,RZ=YR & PX=ZP
the end result is XQ+YR+ZP=1/2P