# The 80-80-20 Triangle

Several triangles due to their specific properties served as a source for research, wonderment, and a variety of problems. Some, like equilateral, right isosceles, golden, Egyptian, 30-60-90, are very well known. One triangle that deserves more recognition has probably entered mathematical folklore some 100 years ago. Tom Rike of Berkley Mathematical Circles mentions its appearance in the Mathematical Gazette, Volume 11 (1922), p. 173. (His online article also provides additional references.) This is the 80-80-20 (or sometimes 20-80-80) triangle, i.e., the isosceles triangle with the apex angle of 20° and the base angles of 80°.

The original problem gave rise to a few modifications; and each of them has been solved in many, many ways. I'll be adding solutions and perhaps problems related to the original one. Any assistance is welcomed.

This page is to serve as the portal for this undertaking.

Original problem | 60-70 variant | Short segments | Long segments | Isosceles partition |

Another partition | 40-50 Variant | 30-40 variant | 20-30 variant | 10-20 variant |

#### Original problem

Let ABC be an isosceles triangle

#### 60-70 Variant

Let ABC be an isosceles triangle

#### Short segments equality

ABC is an isosceles triangle with vertex angle

#### Long segments equality

ABC is an isosceles triangle with vertex angle

#### Partition into adjacent isosceles triangles

The 80-80-20 triangle can be partitioned into isosceles triangle with bases on the legs of the given triangle. For what other apex angles A can isosceles triangle ABC be tessellated with isosceles triangles in a similar manner?

#### Partition that starts with a base angle bisector

The 80-80-20 triangle can be partitioned in a curious way: let BD, with D on AC, bisect angle B. Then

This problem admits an inverse.

#### 40-50 Variant

Let ABC be an isosceles triangle

#### 30-40 Variant

Let ABC be an isosceles triangle

Solutions

#### 20-30 Variant

Let ABC be an isosceles triangle

#### 10-20 Variant

Let ABC be an isosceles triangle

Solutions

### References

- T. Rike, An Intriguing Geometry Problem, Berkeley Math Circle, May 5, 2002.
- H. S. M. Coxeter and S. L. Greitzer,
*Geometry Revisited*, MAA, 1967. - R. Honsberger,
__Four Minor Gems from Geometry__,*Mathematical Gems*II, MAA, 1976. - R. Honsberger,
__Three Solutions to a Variation on an Old Chestnut__,*Mathematical Chestnuts from Around the World*, MAA, 2001. - C. Knop,
__Nine Solutions to One Problem__,*Kvant*, 1993, no 6. - R. Leikin,
__Dividable Triangles - What Are They?__,*Mathematics Teacher*, May 2001, pp. 392-398. - V. V. Prasolov,
*Essays on Numbers and Figures*, MAA, 2000.

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