4. Note that if the two circles have the same radii, then the point, where the line joining the points of intersection meets the line joining the centers, is the midpoint of the line joining the centers. Since we already know that it is the midpoint of the line joining the points of intersection, and that the two lines are perpendicular, we can conclude that if the circles have the same radii, that the two lines are perpendicular bisectors of each other. This gives us a proof that, in a rhombus, the diagonals are perpendicular bisectors of each other. Note that this only gives us this result going in one direction, but the converse will follow easily from the Pythagorean Theorem.

 

Return to text

Analytic Foundations of Geometry

R. S. Wilson