Two systems avoid that feature, so that the coordinates of a point are not affected by moving the triangle, rotating it, or reflecting it as in a mirror, any of which give a congruent triangle, or even by rescaling it to give a similar triangle:
12.
In geometry, "'" Corresponding parts of congruent triangles are congruent " "'( "'CPCTC "') is a succinct statement of a theorem regarding congruent trigonometry, defined as triangles either of which is an isometry of the other.
13.
These are connected as follows : invariants are constant on coinvariants ( for example, congruent triangles have the same perimeter ), while two objects which agree in the value of one invariant may or may not be congruent ( two triangles with the same perimeter need not be congruent ).
14.
This article assumes that a scaling can have a scale factor of 1, so that all congruent shapes are also similar, but some school textbooks specifically exclude congruent triangles from their definition of similar triangles by insisting that the sizes must be different if the triangles are to qualify as similar.
15.
Birkhoff, for instance, uses the axiom which says that " there exists a pair of similar but not congruent triangles . " In any of these systems, removal of the one axiom which is equivalent to the parallel postulate, in whatever form it takes, and leaving all the other axioms intact, produces absolute geometry.
16.
For properties 4 and 5, first observe that any two of the three Johnson circles are interchanged by the reflection in the line connecting " H " and their 2-wise intersection ( or in their congruence, this gives property 5, and also the Johnson circles theorem since congruent triangles have circumscribed circles of equal radius.
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