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Let AOB be a given angle less than 180° and let P be an interior point of the angular region determined by ∠AOB. Show, with proof, how to construct, using only ruler and compasses, a line segment CD passing through P such that C lies on the ray OA and D lies on the ray OB, and CP : PD = 1 : 2.

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Join OP and extend to Q such that OP : PQ = 2 : 1 Draw a line parallel to OA through Q.

It cuts OB at point M (say) take point D on OB such that M is mid point of OD joint DQ and Produce it to meet OA at N. Then by mid-point theorem, Q is mid point of DN. so OQ is median of

Δ ODN

As OP : PQ = 2 : 1,

P is centroid => DP : PC = 2 : 1

Construction of OP : PQ = 2 : 1

obtain mid-point L of OP.

with P as centre and radius PL, arc cuts OP produced at Q

Construction of QM || OA :

With O as centre, draw an arc to cut OQ at S1 and OA at SQ

With Q as centre and same radius, draw an arc to cut OQ at S3

With S3 as centre and radius S1 S2, draw an arc to cut previous arc at S4 join QS4 

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