Proving BE:EX 3:1 in Triangle ABC with Medians

This article explores a geometric proof in triangle ABC, where AD is a median on BC, and BE is a median on AD. If BE is further extended to meet AC at point X, the problem is to prove that BE:EX 3:1.

Step-by-Step Proof

Step 1: Drawing Parallel Lines

First, we draw a line XP parallel to BD. By doing so, two similar triangles are formed which will help us in finding the ratios of the segments.

Step 2: Applying Similarity Theorems

This process involves the following steps:

Triangle BDE is similar to Triangle XPE. This gives us the ratio:

$frac{BD}{PX} frac{DE}{PE}$

Triangle ADC is similar to Triangle APX. This gives us the ratio:

$frac{DC}{PX} frac{AD}{AP}$

Step 3: Using the Given Information

Since BD DC, we have:

$frac{AD}{AP} frac{DE}{PE}$

Multiplying both sides by 2 (as AD 2DE):

$frac{2 times DE}{AP} frac{DE}{PE}$

Simplifying this, we get:

$frac{AP}{PE} 2$

So, simplifying further:

$frac{AP}{PE} - 1 2 - 1$

Step 4: Extending BE to Meet AC at X

We extend BE to meet AC at point X. We draw a line DY parallel to BX. Then, we use the following steps:

In Triangle BEC, we have:

$frac{BD}{DC} frac{XY}{YC}$

From the Basic Proportionality Theorem (BPT),

$BD DC$, hence $frac{XY}{YC} 1$

So, $XY YC$ (Equation 1).

In Triangle ADC, we have:

$AX XY$ (Equation 2).

Step 5: Using Equations to Find AX XY YC

From Equations 1 and 2:

$AX XY YC$ (Equation 5).

Step 6: Proving Similarity of Triangles

We prove the similarity of triangles DYC and BXC by the Angle-Angle (AA) rule:

$frac{DY}{BX} frac{DC}{BC}$

Since $DC frac{1}{2}BC$,

$frac{DY}{BX} frac{1}{2}$

So, $2DY BX$ (Equation 3).

Step 7: Proving Similarity of Triangles AEX and ADY

We prove the similarity of triangles AEX and ADY by the AA rule:

$frac{XE}{DY} frac{AX}{AY}$

From Equation 5, $AX 2AX$;

$frac{XE}{DY} frac{1}{2}$

Therefore,

$2XE DY$ (Equation 4).

Step 8: Concluding the Proof

Combining Equations 3 and 4:

$2 times 2XE BX$

$4XE BX$

So,

$BE 3XE$

Hence,

$frac{BE}{EX} 3 : 1$

Hence Proved