Mathematics

O is any point inside a triangle ABC. The bisector of $$\angle AOB$$, $$\angle BOC$$ and $$\angle COA$$ meet the sides, AB, BC and CA in point D, E and F respectively,
therefore,
$$AD\times BE $$ = $$ DB\times FA$$



ANSWER

False


SOLUTION
Given: $$\triangle ABC$$, CX is bisector of $$\angle C$$,
$$\angle ACY = \angle BCX$$ ...(I)
and $$AX = AY$$

In $$\triangle AXY$$,
$$\angle AXY = \angle AYX$$ (Angles opposite to equal sides)...(II)

Now, $$\angle XYC = \angle AXB = 180$$ (Angles on a straight line)
$$\angle AYX + \angle AYC = \angle AXY + \angle BXY$$
hence, $$\angle AYC = \angle BXY$$....(III)

Also, In $$\triangle AYC$$ and $$\triangle BXC$$
$$\angle AYC + \angle YCA + \angle CAY = \angle BXC + \angle BCX + \angle XBC = 180$$ (Angles sum property)
$$\angle CAY = \angle XBC$$ (From I and III)
or $$\angle CAY = \angle ABC$$
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