In the figure (i) given below, QPX is the bisector of ∠YXZ of the triangle XYZ. Prove that XY : XQ = XP : XZ.

In △XYQ and △XPZ,

∠Q = ∠Z (∵ angles in same segment are equal as arc XY subtends both the angles at the circle.)

∠YXQ = ∠PXZ (∵ QPX is the bisector of ∠YXZ hence it divides the angle in two equal halves.)

△XYQ ~ △XPZ. (By AA axiom)

Since triangles are similar hence, the ratio of corresponding sides are similar,

$\therefore \dfrac{XY}{XP} = \dfrac{XQ}{XZ} \\[1em] \Rightarrow \dfrac{XY}{XQ} = \dfrac{XP}{XZ} \\[1em] \Rightarrow XY : XQ = XP : XZ \\[1em]$

Hence, proved that XY : XQ = XP : XZ.