Geometrical Construction and Plane Geometry
This domain tests accurate compass-and-straightedge construction: bisectors, perpendiculars, parallels, proportional division, regular polygons, tangency, blends and standard conic/locus curves.
Basic constructions
To bisect a line, strike equal arcs from both ends above and below and join the intersections. To bisect an angle, strike an arc from the vertex cutting both arms, then equal arcs from those two points. To drop a perpendicular from a point to a line, use arcs from the point cutting the line and bisect the resulting chord. Parallels are drawn by transferring the perpendicular distance at two points.
Proportional division
To divide a line AB into five equal parts: draw a ray from A at any convenient angle, step off five equal divisions with dividers, join the fifth point to B, then draw lines parallel to that join through the other points. The intercept theorem guarantees equal parts.
Polygons
The radius of a regular hexagon equals its side, so a hexagon is stepped round a circle with the compass set to the radius; a 60° set square gives the same result. A regular pentagon is constructed with the 72° angle at the centre, or by the standard inscribed-pentagon construction.
Tangency and blends
A tangent point always lies on the line joining the two centres. For an external tangent arc of radius R between two circles of radii r1 and r2, set the compass to R − r1 and R − r2 from the respective centres; for an internal (enclosing) tangent use R + r1 and R + r2. The intersection of those arcs is the centre of the blending arc.
Worked example
Blend two circles, radii 20 mm and 30 mm, centres 70 mm apart, with an external arc of radius 50 mm. Strike arcs of 50 − 20 = 30 mm and 50 − 30 = 20 mm from the two centres; their crossing is the blend centre. Draw the arc, then mark the tangent points on the lines joining the blend centre to each circle centre.
Traps
Treating the blend centre as the corner point; forgetting that the tangent point lies on the centre line; leaving heavy construction arcs so the answer is unreadable; drawing an ellipse freehand when the four-centre or concentric-circles method is required.