Proposition I.1 of Euclid's Elements deals with the construction of an equilateral triangle. Propositions I.4, I.8, and I.26 are what we nowadays would call SAS, SSS, ASA theorems, respectively. Triangle is the most basic, simplest of all geometric shapes. It may be argued that circle, not having corners and needing only one quantity to be well defined, may be simpler. Still it's the simplest among all polygons, and, I would speculate that, among all the simplest shapes, triangle offers the greatest variety of forms and definitions. A triangle is of course well defined by its vertices. (By which I mean the relative positions of the vertices. As a set, they may be rotated, translated or reflected - the triangle will remain the same. In contemporary terminology, all such triangles are rather called congruent than equal.)

In general, a triangle is defined by its three elements. SAS, ASA, SSS provide three well known examples. But there is much more. Before listing those that come to mind, let's agree on some notations:

• A, B, C - angles or vertices
• a, b, c - sides opposite to A,B,C, respectively, or their lengths
• h_a, h_b, h_c - altitudes (or sometimes heights) to the sides a,b,c
• m_a, m_b, m_c - medians to the sides a,b,c
• l_a, l_b, l_c - bisectors of the angles A,B,C
• H_a, H_b, H_c - feet of the corresponding heights
• M_a, M_b, M_c - midpoints of sides a,b,c
• L_a, L_b, L_c - feet of the corresponding angle bisectors
• O and R - the center (often circumcenter) and the radius, respectively, of the circumscribed circle. R is known as circumradius, the circle as the circumcircle.
• H - orthocenter, the point of intersection of the three altitudes
• G - center, centroid, barycenter, the point of intersection of the three medians, sometimes also called the median point
• I and r - the center (often incenter) and the radius, respectively, of the inscribed circle. r is known as inradius, the circle as the incircle.
• I_a, I_b, I_c - excenters, i.e., centers of excircles. These are points of concurrency of one internal and two external angle bisectors of ABC
• r_a, r_b, r_c - exradii, i.e., the radii of excircles.
• p - semiperimeter, p = (a + b + c)/2
• aa, bb, cc - straight lines obtained by extending sides a,b,c
• S - area of the triangle
• S_a, S_b, S_c - feet of symmedians AS_a, BS_b, CS_c (Symmedian is the line symmetric to the median in the angular bisector. E.g., AS_a is symmetric to m_a in l_a. In other words, AS_a is isogonal to m_a. The symmedians have many interesting properties.)

An applet below illustrates the geometry of a triangle

Here is the table of triangle constructions. From time to time, I'll be adding constructions to the listed combinations. Every one is welcome to post solutions or solved new combinations to the CTK Exchange. There is no need to log in or sign up for a membership.

1. 9 Point Cirle
2. About a Line and a Triangle
3. The Altitudes
4. The Altitudes and the Euler Line
5. The Angle Bisectors
6. Barycentric Coordinates
7. Bride's Chair, Vecten points
8. Ceva's Theorem
9. The Euler Line and the 9-Point Circle
10. Fagnano's Problem
11. Frieze Patterns
12. The Medians
    • Another Look at Concurrency of the Medians
13. Menelaus Theorem
14. Symmedian and Antiparallel
15. Transitivity in Action
16. Van Obel Theorem and Barycentric coordinates

The many relations that exist between various elements of a triangle are gathered on a separate page. Triangles are classified with respect to the relative sizes of their side lengths and angles.

Copyright © 1996-2008 Alexander Bogomolny