If a line intersects sides AB and AC of a $\triangle ABC$ at D and E respectively and is parallel to BC, prove that $\dfrac{AD}{AB} = \dfrac{AE}{AC}$ (see Fig. 6.13). [Figure: triangle ABC with vertex A at top; line DE drawn parallel to base BC, intersecting AB at D and AC at E.]

Step 1: Apply Basic Proportionality Theorem

Given that the line segment DE is parallel to the side BC, we can apply the Basic Proportionality Theorem (also known as Thales Theorem or Theorem 6.1 in NCERT Class 10 Mathematics, page 1). This theorem states that if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio. Therefore, the ratio of the segments on side AB, namely AD to DB, is equal to the ratio of the segments on side AC, namely AE to EC.

Step 2: Add 1 to both sides

To relate the ratios to the full sides AB and AC, we add 1 to both sides of the equation obtained from the Basic Proportionality Theorem. This is a common algebraic manipulation used to transform segment ratios into full-side ratios.

Step 3: Simplify the fractions

By finding a common denominator and adding, we combine the terms on each side of the equation. On the left side, $\frac{AD}{DB} + 1 = \frac{AD}{DB} + \frac{DB}{DB} = \frac{AD + DB}{DB}$. Similarly, on the right side, $\frac{AE}{EC} + 1 = \frac{AE}{EC} + \frac{EC}{EC} = \frac{AE + EC}{EC}$.

Step 4: Substitute segment sums

From the figure, we observe that the segment AD plus DB constitutes the entire side AB. Similarly, the segment AE plus EC constitutes the entire side AC. Substituting these relationships into the equation from the previous step, we get the ratios involving the full sides AB and AC.

Step 5: Take the reciprocal of both sides

To arrive at the desired form of the proof, we take the reciprocal of both sides of the equation. This operation maintains the equality of the ratios.

Step 6: Rewrite ratios using full sides

We can express DB as $AB - AD$ and EC as $AC - AE$. Substituting these into the equation, we get an expression in terms of AD, AE, AB, and AC.

Step 7: Separate the fractions

By splitting the fractions on both sides, we simplify the equation. For example, $\frac{AB - AD}{AB} = \frac{AB}{AB} - \frac{AD}{AB} = 1 - \frac{AD}{AB}$. Similarly for the right side.

Step 8: Isolate the desired ratio

Subtracting 1 from both sides of the equation and then multiplying by -1 (or simply observing that if $1 - X = 1 - Y$, then $X = Y$), we isolate the ratio we intended to prove. This completes the proof.

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