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By tangent properties,

IL = IJ --- (1)

Also LO = LO --- (2) (obvious)

Also IO and JO are both radii, so IO = JO --- (3)

By (1), (2), and (3), and using SSS postulate, $\triangle LIO \cong \triangle LJO$.

Therefore $\angle ILO = \angle JLO$

$\implies$ LO bisects ∠ILJ