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If Two Sides of a Triangle Are Unequal, Then the Angle Opposite to the Greater Side Is Greater Than Angle Opposite to the Smaller Side

The points ...

Question

The points $D, E$ and $F$ are taken on the sides $A B, B C$ and $C A$ of a $△ A B C$ respectively, such that $D E ∥ A C$ and $F E ∥ A B$ .
Prove that $\frac{A B}{A D} = \frac{A C}{F C}$

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Solution

Given that in $△ A B C, D E ∥ A C$ .
$\frac{B D}{D A} = \frac{B E}{E C}$ (Thales theorem) .... $(1)$
Also, given that $F E ∥ A B$ .
$∴ \frac{B E}{E C} = \frac{A F}{F C}$ (Thales theorem) .... $(2)$
From $(1)$ and $(2)$ , we get
$\frac{B D}{A D} = \frac{A F}{F C}$
$\Rightarrow \frac{B D + A D}{A D} = \frac{A F + F C}{F C}$ (componendo rule)
Thus, $\frac{A B}{A D} = \frac{A C}{F C}$ .

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