The picture shows an isosceles triangle fitting the description.
BO=OC (because O is the midpoint of BC).
BRO=OSC=90º by definition.
Angle ABC=ACB (given, equal angles of triangle ABC).
Angle ROB=SOC=90-ABC=90-ACB (angles in a right-angled triangle).
Triangles ROB and SOC are congruent because all the angles and the hypotenuses are equal.
Therefore OR=OS and BR=SC.
Angle PCB=90-ABC and angle QBC=90-ACB so PCB=QBC.
BC is common to triangles BPC and BQC. Angle BPC=BQC=90º by definition.
Triangles BPC and BQC are congruent (all angles and hypotenuses are equal).
Therefore PC=BQ.