nLab triangle

Idea

The 2-simplex. The polygon with three sides.

Properties

• Pythagorean theorem

 In constructive mathematics

Two of the angles in a triangle on a Euclidean plane are acute angles?.

In classical mathematics, triangles on a Euclidean plane can be characterised as an acute triangle, where the third angle is an acute angle?, a right triangle, where the third angle is a right angle, or an obtuse triangle, where the third angle is an obtuse angle?

However, in constructive mathematics, that triangles on a Euclidean plane are either acute, right, or obtuse is equivalent to the analytic LPO, since it requires that

$$\gamma \lt 90^\circ \vee \gamma = 90^\circ \vee \gamma \gt 90^\circ$$

be true for a real-valued angle measure $\gamma$, which is equivalent to the analytic LPO for a set of real numbers $\mathbb{R}$ by subtracting $90^\circ$ from both sides of each inequality. This means that without the analytic LPO, there are triangles where it is not possible to decide whether the triangle is acute, right, or obtuse.

Related concepts

• trigonometry

• circle

• square

• tetrahedron

• triangle type?

Reference

• Wikipedia, Triangle