Given: triangle ABC. Point P lies on the side AB of the triangle, O-lies on the segment CP so that AO = OB.

Given:
triangle ABC,
point P lies on the side AB,
point O lies on the segment CP,
AO = ОВ,
∠AOС = ∠ВOС,
РВ = 1/3 АС,
AR = 3 cm.
Find the perimeter P ABC -?
Decision:
1) Consider the triangle AOB. It is isosceles, since ВO = AO. Then PO is the median. Therefore AR = PB = 3 cm;
2) Since PB = 1/3 AC;
AC = 3 * PB;
AC = 3 * 3;
AC = 9 cm;
3) Consider the triangles AOC and COB. they have ВO = AO, ∠AOС = ∠ВOС and the OС side is common. On two sides and the angle between them, the triangles are AOC = COB. Then AC = BC = 9 cm;
4) P ABC = 6 + 9 + 9;
R ABC = 24 (cm).
Answer: 24 centimeters.