Geometry - Construct the segment $x=\sqrt{a^2+ab+b^2}$, if $a$ and $b$ are given segments.

Question

I have a difficult geometry problem with constructing segments. The problem is:

Construct the segment $x=\sqrt{a^2+ab+b^2}$, if $a$ and $b$ are given segments.

Can anyone help me?

Answer 1

• $x$ is edge length of the triangle with two edges lengths $a,b$ and internal angle $120$ degrees at vertice intersection of the two edges
• from cosines law, length of the third edge verifies $x^2 = a^2 + ab + b^2$
• so you need to construct $120$ degrees angle: how to do that? draw a circle and split into three congruent sectors (with a compass, making a circumscribed hexagon, you can make it easily)
• now draw to two radial segments (lengths a and b) from the center; join the two radial segments endpoints to get a segment length $x$