Side-Side-Side Triangle Method for Triangle Construction and Duplication Given the length of three side of a triangle ABC, we can construct an isometric triangles DEF and DFH with the use of a straight lines, or ruler and compass. The construction method is described in greater detail below. Click on image SSS method   Given the length of 3 sides of an existing triangle, we can use ruler and compass (or strings) to construct an  isometric copy or two.  Method. Step 1.  Draw a base DF equal in length to the larges side AC of the original triangle.  Step 2.  Draw circles with radii given by the lengths of the other two sides in triangle ABC, centred at end-points D and F of line segment DF. Step 3.  use the intersection points of the circles, here labeled E and F to provide a third vertex of two triangles with side EF. Then the two resulting triangles are isometric to the original triangle  Empirical Remark 1: Drawings with the larger radius circle centered at D instead of F would result in two more triangles isometric to triangle ABC. Thus for any line segment DF, there are upto four instances of the SSS triangle construction method. Empirical Remark 2:  The foregoing construction method can be applied starting with side AC of the triangle ABC. That construction method with it four variations would result in 4 triangles isometric to ABC.   The first would be coincident with ABC. A second would have a vertex B' on the same side of AC as B.  A third would have a vertex D given by the reflection of B across side AC, or the line through it. The fourth would have a vertex D' equal to the reflection of B' across side AC or the line through it.  The triangle AD'C would be given by a 180 degree rotation of the triangle ABC about the center of side AC.   SSS Isometry Assumption If there is a correspondence between the vertices such that corresponding sides are have equal lengths then the triangles are isometric. Moreover,  both could be built or rebuilt by the same construction SSS method, and each may be moved by a sequence of  translations, rotations and/or reflections, so that sides and vertices coincide with the other.

A side-side-side like Construction - Location of a point.

Scenario 1:  A point P is to be placed 20 meters away from a point A and 25 meters away from a point B, and north of both points.   Two surveyors attach a string (or tape measures) of length 20 meters at A and another of length 25 meters at B. Holding the strings taught they move the other ends of the string until the ends meet. That locates the point P. 

Scenario 2.  Two surveyors observe a point P is 20 meters away from a point A, and 25 meters away from a point B, and further north than both.  On a map drawn to a scale of 1 cm to 10 meter, they locate P on the map by finding the Northern most intersection of a circle of radius 2.0 cm centered at the map location of A with another circle of radius 2.5 cm centered at the map location of B. That yields the coordinates of P on the map and actual.

Uniqueness Observation and Assumption: If one side AB of a triangle ABC is given then the Side Side Side construction method applied to end points of the line segment AC leads to two isometric triangles, the original triangle ABC and a second triangle ADC - isometric by the assumption above.

The discussion of this situation will be continued below.

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