Euler made contributions to the new field of graph theory including results that determine the traceability of a graph. He used this to "solve" the Konigsberg bridge problem.
Euler also made contributions to what would become topology. His formula V-E+F=2 relates the number of faces, vertices, and edges of a convex polyhedron giving rise to its Euler characteristic.
For a triangle the circumcenter (the center of the circumscribed circle), the orthocenter (the point where the perpendicular bisectors of the sides meet) and the centroid (the point where the medians meet) all lie on what is called the Euler line.
Leonhard Euler was a Swiss mathematician and was one of the most influential mathematicians of the 18th century. He is most famous for introducing the concept of the math function f(x) and for what is now known as Euler's number e, which is the base of the natural logarithm.
In terms of geometry, Euler developed the proof relating the radii of the circumcircle and incircle for a triangle. He also developed Euler's formula:
e^(ix) = cos(x) + i*sin(x)
This establishes the relationship between the complex exponential function (e^(ix)) and trigonometric functions (sine and cosine). Geometrically, this defines the unit circle on the complex number plane where x is the angle between a line traversing from the origin to a point on the unit circle and the positive real axis. Please see the links below for further information and diagrams, particularly the first link for the unit circle diagram.