Let  If  then the set of congruence classes of triangles (in the Euclidean plane) that have area Mand (at least) one side of length ais in one-to-one correspondence with  The following assertions hold “up to congruence”. There is only one non-right (Euclidean) triangle with area  and sides of length aand  its other side has length  There is only one triangle with area  and (at least) two sides of length a. If a triangle has area  and (at least) two sides of length d, then  If  there are exactly two (non-congruent) triangles with area  and (at least) two sides of length d. If  then there are exactly two (pairwise non-congruent) triangles with area  and sides of length aand  The only prerequisite for this note is elementary analytic geometry.

Euclidean geometry, triangle, area, right triangle, isosceles triangle, Heron’s formula, distance formula, circle, line.