An auxiliary line (or helping line) is an extra
line
Line most often refers to:
* Line (geometry), object with zero thickness and curvature that stretches to infinity
* Telephone line, a single-user circuit on a telephone communication system
Line, lines, The Line, or LINE may also refer to:
Arts ...
needed to complete a
proof
Proof most often refers to:
* Proof (truth), argument or sufficient evidence for the truth of a proposition
* Alcohol proof, a measure of an alcoholic drink's strength
Proof may also refer to:
Mathematics and formal logic
* Formal proof, a con ...
in
plane geometry. Other common auxiliary constructs in elementary plane synthetic geometry are the helping circles.
As an example, a proof of the theorem on the
sum of angles of a triangle
In a Euclidean space, the sum of angles of a triangle equals the straight angle (180 degrees, radians, two right angles, or a half- turn).
A triangle has three angles, one at each vertex, bounded by a pair of adjacent sides.
It was unknown ...
can be done by adding a straight line parallel to one of the triangle sides (passing through the opposite vertex).
Although the adding of auxiliary constructs can often make a problem obvious, it's not at all obvious to discover the helpful construct among all the possibilities, and for this reason many prefer to use more systematic methods for the solution of geometric problems (such as the coordinate method, which requires much less ingenuity).
References
External links
*http://www.cut-the-knot.org/Generalization/MenelausByEinstein.shtml On Einstein's opinion regarding proofs that use the introduction of additional constructs
Geometry
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