When two chords intersect, each of them is divided into two-line segments. Show that if the intersecting chords are of equal length, then the line segments of one chord are equal to the corresponding line segments of the other chord.

Let two chords AB and CD of a circle with centre O intersect at a point P inside the circle.

Given: AB = CD ...(i)

Since, the perpendicular distances from the centre are equal.

i.e., OM = ON

Consider triangles OMP and ONP,

OP = OP [Common sides]

OM = ON [Proved above]

∠OMP = ∠ONP = 90°

By RHS congruence, ΔOMP ≅ ΔONP.

Hence, MP = NP [By CPCT] ...(ii)

Since, M is the midpoint of AB

$$AM = BM = \frac{AB}{2} \quad \text{...(iii)}$$

$$\text{and} \quad CN = DN = \frac{CD}{2} \quad \text{...(iv)}$$

From (i), (iii) and (iv),

$$AM = CN \quad \text{...(v)}$$

$$\text{and} \quad BM = DN \quad \text{...(vi)}$$

Adding (ii) and (vi),

$$MP + BM = NP + DN$$

$$BP = DP$$

Subtract (ii) from (v),

$$AM - PM = CN - PN$$

$$AP = CP$$

Hence, proved.