A.6.3.RT.OE:5.1 - Keep a segment available in different geometric groupings
The subject task is to construct an equilateral triangle on a nonzero segment AB in the Euclidean plane. The geometric Method permits a circle centered at A through B, another centered at B through A, an intersection C and the joins AC and BC. The subject premises and construction justify those objects, including the intersection.
Prepare a diagram in which the same A, B and C retain their labels and the segments can be examined in these groupings:
| Examination | Relations available |
|---|---|
| Circle centered at A | AB and AC are radii, so AB = AC. |
| Circle centered at B | BA and BC are radii, so BA = BC. |
| Triangle ABC | AB, BC and CA are its three sides. |
Keep the shared segment visible while moving between the first two groupings. AB and BA denote the same undirected segment; their lengths are equal. The subject argument can therefore combine the radius equalities and conclude that the triangle’s three sides are equal.
The expression’s contribution is the jointly usable arrangement of those parts. Geometry supplies the constructions, radius property and equality reasoning. Measuring the drawing is unnecessary for that argument. If a later construction needs one of these segments extended, retain the point labels and the circle relations that justify its earlier length.
This develops Macbeth’s analysis of Euclid I.1.