# The many ways to construct a triangle

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 ΔABC - 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 CutTheKnotMath facebook page. There is no need to log in or sign up for a membership.

(A. S. Posamentier's *Advanced Euclidean Geometry* lists 179 triangle construction problems (p. 178-180), but solves just a dozen of selected ones.)

### More about remarkable points, lines and identities in a triangle

- 9 Point Cirle
- About a Line and a Triangle
- The Altitudes
- The Altitudes and the Euler Line
- The Angle Bisectors
- Barycentric Coordinates
- Bevan's Point and Theorem
- Bride's Chair, Vecten points
- Ceva's Theorem
- The Euler Line and the 9-Point Circle
- Fagnano's Problem
- Fermat point and 9-point Centers
- Frieze Patterns
- Gergonne and Soddy Lines Are Perpendicular
- The Medians
- Menelaus Theorem
- Symmedian and Antiparallel
- Transitivity in Action
- 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.

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