Problem
If the bisectors $OM$ and $ON$ of the tangent angles $∠AOB$ and $∠BOC$ are perpendicular to each other, $OA$ and $OC$ form a straight line.
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Solution
First, from the problem statement,
$$∠AOM=∠MOB.$$
$$∠NOB=∠NOC.$$
By assumption,
$$∠MON=∠R,$$
$$∴ \ ∠MOB+∠NOB=∠R,$$
$$∴ \ (∠AOM+∠MOB)+(∠NOB+∠NOC)=2∠R,$$
$$∴ \ ∠AOB+∠BOC=2∠R,$$
$$∴ \ ∠AOC=2∠R.$$
Therefore, $OA$ and $OC$ form a straight line.
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Reference
Teiichiro Sasabe (1976) The Encyclopedia of Geometry (2nd edition), Seikyo-Shinsha, p.6