A geometry puzzle centered around the triangle or the centers of a triangle.

Triangles are one of the simplest geometric shapes. Yet, their combinatorics is quite non-trivial and often leads to nice puzzles. The following list summarizes some central types of special triangles:

  • An isosceles triangle is a triangle with (at least) two equal sides. Equivalently, an isosceles triangle is a triangle with two equal angles.
  • An equilateral triangle is a triangle with all three sides equal. Equivalently, an equilateral triangle is a triangle with three angles of 60 degree.
  • A right-angled triangle is a triangle that has a right angle (90 degree) in it.
  • An obtuse triangle is a triangle that has one angle strictly larger than 90 degree.
  • An acute triangle is a triangle that has all angles strictly smaller than 90 degree.

Furthermore, there are two important relations between pairs of triangles:

  • Two triangles are similar, if they have the same three angles.
  • Two triangles are congruent, if they have the same three side lengths.

It is easy to see that congruent triangles are always similar to each other, whereas similar triangles are not necessarily congruent.

In puzzles, one of the most-used properties of a triangle is the triangle inequality: If the three side lengths of a triangle are a<=b<=c, then we always have a+b>=c. The case with a+b=c yields a degenerate triangle whose area is zero.