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Euclidean Geometry |
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Complex Numbers
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19-August-2008
Kite Construction -Duplication of a Triangle using One of its sides.
Definition: A kite is a quadrilateral so that in two pairs of adjacent sides, the sides are equal in length.
1. Side-Side-Side Method: If one side AB of a triangle ABC is given then variations 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.
2. Angle-Side-Angle Method: 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 ASA method.
3. Side-Angle-Side Method: In triangle ABC, if the measure of angle CAB, and the measure of side AB is given then variants of the Side-Angle-Side construction method applied to end points of the line segment AC (AC being the leads to two isometric triangles, the original triangle ABC and a second triangle ADC - isometric by ASA method.
4. Reflection Across a side AC Method
Drop a perpendicular from the vertex B to the line segment AC or the line AC. That gives a point E. On the line BE find a point D on the "non-B side" of E so that BE and DE have equal lengths. The SAS isometry criteria triangles BED and DEA are isometric, and triangles BEC and triangles DEC are also isometric. Thus triangle ABC and ADC are isometric via the SSS criteria.
Summary: All four triangle construction methods lead to a triangle ADC isometric to the original triangle ABC.
Assumption: The point D is unique.
Properties of AC and BD.
Triangle ABD is isosceles as sides AD and AB have equal lengths.. Therefore angles ABD and Angles ADB are equal in measure.
In triangles EAB and EDA, sides AE is common; sides AB and AD have the same length since triangles ACB and ADB are isometric; and angles BAE and DAE are equal, agains since triangles ABC and ADC are isometric. Thus triangles EAC and EDA are isometric by the SAS isometry criteria. That implies E is equidistant to points B and D, and that BD is perpendicular to AB as angles BEA and DEA are equal and their sum is 180 degrees (two right angles).
Thus D is the reflection of B across the line segment and side AB.
The quadrilateral ABCD is a kite.