Q. 95.0( 1 Vote )

In Δ ABC, such that . Prove that X, Y, Z are the mid-points of , , respectively.

Given: In ∆ABC, XBC, YCA, ZAB.

Also, XY||AB, YZ||BC and ZX||AC.

To prove: X,Y and Z are the mid-points of BC, CA and AB respectively.

Proof: In ∆ABC, YZ||BC.

…(1)

In ∆ABC, XY||AB.

…(2)

From (1) and (2),

…(3)

In ∆ABC, ZX||AC.

…(4)

From (3) and (4),

AZ2 = BZ2

AZ = BZ

Hence, A-Z-B and AZ = BZ.

Thus, Z is the mid-point of AB.

Similarly, Y is the mid-point of AC and X is the midpoint of BC.

Therefore, X, Y and Z are the mid-points of BC, CA and AB respectively.

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