|
Euclidean Geometry |
|
Complex Numbers
|
19-August-2008
Parallelogram Construction Methods Based on Triangles
Construct a quadrilateral with opposite equal by rotation of 180 degrees about the midpoint of one side of a triangle. The side becomes a diagonal.
Consequence: Each triangle construction method defines a triangle and hence a parallelogram.
We consider the SAS, SSS and ASA methods for constructing parallelograms.
Side Angle Side Method:
Construction of parallelogram from two sides and an angle between them:
Initial Data:
Step 1: Complete a triangle:
Step 2: Draw a line parallel to one of the original sides:
Step 3: Draw a line parallel to the other of the original sides:
The resulting diagram gives a figure with opposite sides of equal length due to the ASA isometry criteria. Thus the resulting figure is a parallelogram:
Remark: The same result and same arguments apply if the triangle in step 1 is rotated 180 degrees about the mid-point of the side drawn to complete the triangle. |
Alternate Step 2 & 3. Employ the Side-Side-Side
triangle construction method to construct a quadrilateral with
opposite sides of equal length.
The resulting quadrilateral is a parallelogram. Second Alternate Steps 2 & 3. Employ the Side-Angle-Side triangle construction method to construct a quadrilateral with one pair of sides parallel and equal in length.
The resulting quadrilateral is a parallelogram. Remark: The same result and same arguments above apply if the triangle in step 1 is rotated 180 degrees about the mid-point of the side drawn to complete the triangle. |
| It would seem that all three or four construction methods above give the same parallelogram with four vertices all the same location. | |
Unique 4th vertex assumption: The location of three vertices of a parallelogram uniquely determine the location of the fourth vertex.
|
|
 |
Equivalently, the specification of two adjacent sides of a parallelogram uniquely determine the other two sides and fourth vertex. Observe the end points of the adjacent side provide three of the four vertices.
A look ahead: The latter assumption will be employed later in a proof of the distributive law for rotations over vector addition.