D is the mid-point of side BC of a Δ ABC. AD is bisected at the point E and BE produced cuts AC at the point X. Prove that BE: EX = 3 : 1.

Given: In ΔABC, D is the midpoint of BC and E is the midpoint of AD.

To prove: BE: EX = 3: 1

Theorem Used:

1.) If two corresponding angles of two triangles are equal, the triangles are said to be similar.

2.) Basic proportionality theorem:

If a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides the two sides into same ratio.

Proof:

Const: Through D, Draw DF||BX

In ΔEAX and Δ ADF

∠EAX = ∠ADF (Common)

∠AXE = ∠DAF (Corresponding angles)

Then, ΔEAX ~ Δ ADF

So,

(Corresponding parts of similar triangle are proportion)

(AE = ED given)

⇒ DF = 2EX. ……………(i)

In ΔCDF and ΔCBX (By AA similarity)

SO, (Corresponding parts of similar triangle area proportion)

(BD = DC given)

⇒BE + EX = 2DF

⇒BE = EX = 4EX

⇒BE = 4EX – EX

⇒BE = 4EX – EX

⇒BE = 3EX