Lines $$l$$ and $$m$$ intersect in O. Explain how you will construct a triangle OPQ such that $$P \in l$$, $$Q \in m$$, $$\overline{OP}$$ and $$\overline{OQ}$$ are equal in length and $$\overline{PQ}$$ is of given length $$a$$.

PLACEHOLDER ANSWER - the answer is a construction. Let $$\theta$$ be the angle between $$l$$ and $$m$$ at O; triangle OPQ must be isosceles with apex angle $$\theta$$, so its base angles are each $$\frac{180^\circ - \theta}{2}$$. Draw a segment $$XY$$ of length $$a$$ anywhere, construct angles of $$\frac{180^\circ - \theta}{2}$$ at X and Y on the same side, and let the arms meet at Z; then $$ZX = ZY = s$$ is the required equal side. Finally, with centre O and radius $$s$$, mark P on $$l$$ and Q on $$m$$; triangle OPQ is congruent to $$ZXY$$, so $$OP = OQ$$ and $$PQ = a$$.