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
  • ha, hb, hc - altitudes (or sometimes heights) to the sides a,b,c
  • ma, mb, mc - medians to the sides a,b,c
  • la, lb, lc - bisectors of the angles A,B,C
  • Ha, Hb, Hc - feet of the corresponding heights
  • Ma, Mb, Mc - midpoints of sides a,b,c
  • La, Lb, Lc - 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.
  • Ia, Ib, Ic - excenters, i.e., centers of excircles. These are points of concurrency of one internal and two external angle bisectors of ΔABC
  • ra, rb, rc - 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
  • Sa, Sb, Sc - feet of symmedians ASa, BSb, CSc (Symmedian is the line symmetric to the median in the angular bisector. E.g., ASa is symmetric to ma in la. In other words, ASa is isogonal to ma. The symmedians have many interesting properties.)

An applet below illustrates the geometry of a triangle


This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


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.)

a, b, C (SAS) A, B, c (ASA) a, b, c (SSS) A, a, b (ASS)
Ma, Mb, Mc a, b, mc a, b, mb ma, mb, c
ma, mb, b Ha, Hb, Hc hc, lc, mc R, a, b
R, ha, a R, ma, a ha, b, c ha, hb, b
ha, hb, c ha, a, b ma, mb, hc ha, hb, mc
A, hb, hc a, hb, R ha, hb, ma A, ha, ma
a, b, lc A, ha, p A, R, r a, R, r
aa, Hb, Hc ha, hb, hc A, a, ha A, a, ma
a, hb, lc A, B, hc A, ha, la A, a, r
A, a, R A, B, p a, b, A A, B, lc
ma, ha, mb a, ha, ma a, ha, mb a, hb, ma
a, hb, mb a, hb, mc A, ha, hb ma, mb, mc
la, lb, lc a, la, ha A, O, H A, B, G
a, ma, la A, B, H A, B, I O, H, I
ma, ha, hb ma, hb, hc ma, ha, la R, a, ma
A,a,b+c A,b,a+c A,a,b-c ma, mb, a/b
R, a, mb A, a, la ha, la, b A, mb, ha
A, r, ma a, A, mc/mb a, r, and ha A, r, and c-a
A, r, ha la, ha, R la, ha, r ma, ha, R
mb, ha, A mb, R, A ha, ma, r aa, bb, the Euler line
A, O, I R, r, hA

More about remarkable points, lines and identities in a triangle

  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. Bevan's Point and Theorem
  8. Bride's Chair, Vecten points
  9. Ceva's Theorem
  10. The Euler Line and the 9-Point Circle
  11. Fagnano's Problem
  12. Fermat point and 9-point Centers
  13. Frieze Patterns
  14. Gergonne and Soddy Lines Are Perpendicular
  15. The Medians
  16. Menelaus Theorem
  17. Symmedian and Antiparallel
  18. Transitivity in Action
  19. 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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