Perpendicular bisector

The perpendicular bisector b of a line segment AB is a line which is perpendicular to the line segment and passes from the middle of it.



We can see that the distance of any point of the perpendicular bisector is equal from the two endpoints of the line segment. AC = BC.

The proof is simple. We compare the two triangles ADC and BDC.

The equality of the distances AC and BC is used in many other proofs in euclidean geometry.

The reverse is also true.

Any point which has equal distances from the two endpoints of a line segment is on the perpendicular bisector of it.

Bisectors of adjacent supplementary angles are perpendicular

A simple property of bisectors of supplementary angles is that they are perpendicular.
This is a very simple to prove but impressive theorem. If we take the half of suplementary angles it will be the half of $180^o$.

Proposition I

The first proposition in Euclid's Elements is the construction of an equilateral triangle with compass and straightedge.
"On a given finite straight line to construct an equilateral triangle."




The side AB is the radius of the circles BCD and ACE.
The side AC is the radius of the circle BCD.
The side BC is the radius of the circle ACE.
Therefore the line segments AB, AC and BC are equal.
Therefore the triangle is equilateral.
It is a simple and beautiful construction.