Loci in construction problems: a point, a circle, a triangle
Find the wanted point as the crossing of two loci. Three tasks from the Ukrainian book: a point on a line at a given distance from another line, a circle tangent to a line at a given point, and a triangle from a side and the circumradius.
- Ukraine: grade 7
Guide
Construction problems are solved with a compass and an unmarked ruler in four steps: analysis, construction, proof, investigation. The locus method: describe the wanted point by two conditions, draw the two loci and take their crossing. (This is Ukrainian Geometry 7, §24; in England it is GCSE locus and construction work.)
Task: on the line m find a point at the distance a from the line AB. All points at the distance a from AB lie on two lines parallel to AB; the wanted point is a common point of m and these lines. Investigation: two solutions, one, none or infinitely many (when m lies along one of the parallels).
Task: construct a circle through B tangent to the line m at A. The centre O lies on the perpendicular to m at A (the radius to the point of contact is perpendicular to the tangent) and on the perpendicular bisector of AB (OA = OB).
Task: construct a triangle ABC from the side AB and the circumradius R. The centre O is at the distance R from A and from B, so it is a crossing of two circles of radius R about A and B. Any point C of the circle (O; R) gives a triangle. If AB < 2R there are infinitely many, if AB > 2R none.
After every construction ask: how many solutions can the problem have?