5-Con triangles

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The smallest 5-Con triangles with integral sides. 5-Con-triangles-8-12-18-27.svg
The smallest 5-Con triangles with integral sides.

In geometry, two triangles are said to be 5-Con or almost congruent if they are not congruent triangles but they are similar triangles and share two side lengths (of non-corresponding sides). The 5-Con triangles are important examples for understanding the solution of triangles. Indeed, knowing three angles and two sides (but not their sequence) is not enough to determine a triangle up to congruence. A triangle is said to be 5-Con capable if there is another triangle which is almost congruent to it.

Contents

The 5-Con triangles have been discussed by Pawley:, [1] and later by Jones and Peterson. [2] They are briefly mentioned by Martin Gardner in his book Mathematical Circus. Another reference is the following exercise [3]

Explain how two triangles can have five parts (sides, angles) of one triangle congruent to five parts of the other triangle, but not be congruent triangles.

A similar exercise dates back to 1955, [4] and there an earlier reference is mentioned. It is however not possible to date the first occurrence of such standard exercises about triangles.

Examples

There are infinitely many pairs of 5-Con triangles, even up to scaling.

Results

5-Con triangles with the same greatest side. 5-Con-triangles-fixed-greatest-side.svg
5-Con triangles with the same greatest side.

Further remarks

Two 7-Con quadrilaterals. 7-Con-quadrilateral.svg
Two 7-Con quadrilaterals.

Defining almost congruent triangles gives a binary relation on the set of triangles. This relation is clearly not reflexive, but it is symmetric. It is not transitive: As a counterexample, consider the three triangles with side lengths (8;12;18), (12;18;27), and (18;27;40.5).

There are infinite sequences of triangles such that any two subsequent terms are 5-Con triangles. It is easy to construct such a sequence from any 5-Con capable triangle: To get an ascending (respectively, descending) sequence, keep the two greatest (respectively, smallest) side lengths and simply choose a third greater (respectively, smaller) side length to obtain a similar triangle. One may easily arrange the triangles in the sequence in a neat way, for example in a spiral. [1]

One generalization is considering 7-Con quadrilaterals, i.e. non-congruent (and not necessarily similar) quadrilaterals where four angles and three sides coincide or, more generally, (2n-1)-Con n-gons. [1]

References

  1. 1 2 3 Pawley, Richard G. (1967). "5-Con triangles". The Mathematics Teacher. 60 (5, May 1967). National Council of Teachers of Mathematics: 438–443. doi:10.5951/MT.60.5.0438. JSTOR   27957592.
  2. Jones, Robert T.; Peterson, Bruce B. (1974). "Almost Congruent Triangles". Mathematics Magazine. 47 (4, Sep. 1974). Mathematical Association of America: 180–189. doi:10.1080/0025570X.1974.11976393. JSTOR   2689207.
  3. School Mathematics Study Group. (1960). Mathematics for high school--Geometry. Student's text. Geometry. Vol. 2. New Haven: Yale University Press. p. 382.
  4. Thebault, Victor; Pinzka, C. F. (1955). "E1162". The American Mathematical Monthly. 62 (10). Mathematical Association of America: 729–730. doi:10.1080/00029890.1955.11988730. JSTOR   2307084.
  5. Buchholz, R. H.; MacDougall, J. A. (1999). "Heron Quadrilaterals with sides in Arithmetic or Geometric progression". Bulletin of the Australian Mathematical Society. 59 (2): 263–269. doi: 10.1017/s0004972700032883 . hdl: 1959.13/803798 .