Ruler and Compass Construction to Bisect an Angle.
Step 1: Draw a circle or arc centered at the angle vertex which intersect both rays.
It is not necessary to draw all the circle. The intersection points are equidistant from the vertex since the latter is the center of the circle.
Next draw circles of the same radii centered at the intersection points. Those circles will have two intersection points, one at the angle vertex and one on the other side of the line segmenet joining them.
Step 3: Draw line segment or ray AD.
That completes the construction.
Claim: The line segment or ray AD bisects angle BAC.
Proof of Claim:
The points A, B, C and D provided the endpoints of several line segments and vertices of several triangles as indicated below.
By construction line segments AB, AC, BD and CD have the same length r given by the radius of the first drawn circle. Now triangles ABD and ACD are isometric by the SSS criteria. Thus angle BAD and DAC are equal. Hence line segment AD and the ray through it, bisect angle BAC.
Euclidean Geometry
with a geometry based
based development of
complex numbers
24 Lessons:
Easy Consequences of this (newest) Complex Number. Starter Lesson in this site folder follow below.