He also discovered the law of sines for spherical triangles:

To finish the proof by mathematical induction, it remains to show that the theorem is true for triangles.

Isoperimetric points in the sense of Veldkamp exist only for triangles satisfying certain conditions.

However, the incenter lies on the Euler line only for isosceles triangles.

These three new triangles should now have two sides that are available for further triangles to be drawn.

This idea calls for right triangles at a ratio of five to four to three used in the gateways to measure all parts.

Heron's formula as given above is numerically unstable for triangles with a very small angle.

There was no end to the additions that could be made; there was always room for yet smaller triangles.

Another part of the formalism was low regard for triangles which were dismissed as ancient, pagan, or Christian.

Thus for acute and right triangles the feet of the altitudes all fall on the triangle.