Mathematics
In triangle ABC, D and E are points on side AB such that AD = DE = EB. Through D and E, lines are drawn parallel to BC which meet side AC at points F and G respectively. Through F and G, lines are drawn parallel to AB which meet side BC at points M and N respectively. Prove that :
BM = MN = NC.
Mid-point Theorem
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Answer
By mid-point theorem,
The line segment joining the mid-points of any two sides of a triangle is parallel to the third side and is equal to half of it.
By converse of mid-point theorem,
The straight line drawn through the mid-point of one side of a triangle parallel to another, bisects the third side.
By equal intercept theorem,
If a transversal makes equal intercepts on three or more parallel lines, then any other line cutting them will also make equal intercepts.
In △ AEG,
D is the mid-point of AE and DF || EG.
∴ F is mid-point of AG (By converse of mid-point theorem)
∴ AF = FG ………(1)
Since,
DF || EG || BC and DE || BE
∴ FG = GC [By equal intercept theorem]………..(2)
From equation (1) and (2), we get :
⇒ AF = FG = GC
Since, AB || FM || GN and AF = FG = GC
∴ BM = MN = NC [By equal intercept theorem]
Hence, proved that BM = MN = NC.
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