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Ratio of Area of Similar Triangles

O is any poin...

Question

O is any point inside a triangle ABC. The bisector of $∠$ AOB, $∠$ BOC and $∠$ COA meet the sides AB. BC and CA in point D. E and F respectively. Show that AD $\times$ BE $\times$ CF = DB $\times$ EC $\times$ FA. [4 MARKS]

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Solution

Concept : 1 Mark
Application : 1 Mark
Proof : 2 Marks

In $Δ A O B, O D$ is the bisector of $∠ A O B$ .

$∴$ $\frac{O A}{O B}$ = $\frac{A D}{D B} . . . . . . . . . . . . . (1)$

In $Δ B O C, O E$ is the bisector of $∠ B O C$ .

$∴$ $\frac{O B}{O C}$ = $\frac{B E}{E C} . . . . . . . . . . . . . (2)$

In $Δ$ COA, OF is the bisector of $∠$ COA.

$∴$ $\frac{O C}{O A}$ = $\frac{C F}{F A} . . . . . . . . . . . . . (3)$

Multiplying the corresponding sides of (1), (2) and (3), we get

$\frac{O A}{O B}$ $\times$ $\frac{O B}{O C}$ $\times$ $\frac{O C}{O A}$ = $\frac{A D}{D B}$ $\times$ $\frac{B E}{E C}$ $\times$ $\frac{C F}{F A}$

$\Rightarrow$ 1 = $\frac{A D}{D B}$ $\times$ $\frac{B E}{E C}$ $\times$ $\frac{C F}{F A}$

$\Rightarrow$ DB $\times$ EC $\times$ FA = AD $\times$ BE $\times$ CF

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O is any point inside a triangle ABC. The bisector of ∠AOB, ∠BOC and ∠COA meet the sides AB. BC and CA in point D. E and F respectively. Show that AD × BE × CF = DB × EC × FA. [4 MARKS]

O is any point inside a triangle ABC. The bisector of $∠$ AOB, $∠$ BOC and $∠$ COA meet the sides AB. BC and CA in point D. E and F respectively. Show that AD $\times$ BE $\times$ CF = DB $\times$ EC $\times$ FA. [4 MARKS]