Category: Science
Question: The Euler (“OIL-er”) line passes through the orthocenter (“OR-tho-SEN-ter”), circumcenter, and centroid (“SEN-troyd”) in this figure. The area of this polygon can be found by halving the product of two sides and the sine of the angle opposite of the third side or by simply using Heron's formula. It is the only polygon that cannot be concave, because its angles must add to 180 degrees. For 10 points, name this polygon with three sides.
Answer: {triangle}

A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called vertices, are zero-dimensional points while the sides connecting them, also called edges, are one-dimensional line segments. The triangle's interior is a two-dimensional region. Sometimes an arbitrary edge is chosen to be the base, in which case the opposite vertex is called the apex.
In Euclidean geometry, any two points determine a unique line segment situated within a unique straight line, and any three points, when non-collinear, determine a unique triangle situated within a unique flat plane. More generally, several points in Euclidean space of arbitrary dimension determine a simplex.
In non-Euclidean geometries three straight segments also determine a triangle, for instance a spherical triangle or hyperbolic triangle. A geodesic triangle is a region of a general two-dimensional surface enclosed by three sides which are straight relative to the surface. A curvilinear triangle is a shape with three curved sides, for instance a circular triangle with circular-arc sides. This article is about straight-sided triangles in Euclidean geometry, except where otherwise noted.
A triangle with vertices 
  
    
      
        A
        ,
      
    
    {\displaystyle A,}
  
 
  
    
      
        B
        ,
      
    
    {\displaystyle B,}
  
 and 
  
    
      
        C
      
    
    {\displaystyle C}
  
 is denoted 
  
    
      
        △
        A
        B
        C
        .
      
    
    {\displaystyle \triangle ABC.}
  
 In describing metrical relations within a triangle, it is common to represent the length of the edge opposite each vertex using a lower-case letter, letting 
  
    
      
        a
      
    
    {\displaystyle a}
  
 be the length of the edge 
  
    
      
        B
        C
        ,
      
    
    {\displaystyle BC,}
  
 
  
    
      
        b
      
    
    {\displaystyle b}
  
 the length of 
  
    
      
        C
        A
        ,
      
    
    {\displaystyle CA,}
  
 and 
  
    
      
        c
      
    
    {\displaystyle c}
  
 the length of 
  
    
      
        A
        B
      
    
    {\displaystyle AB}
  
; and to represent the angle measure at each corner using a Greek letter, letting 
  
    
      
        α
      
    
    {\displaystyle \alpha }
  
 be the measure of angle 
  
    
      
        ∠
        C
        A
        B
        ,
      
    
    {\displaystyle \angle CAB,}
  
 
  
    
      
        β
      
    
    {\displaystyle \beta }
  
 the measure of 
  
    
      
        ∠
        A
        B
        C
        ,
      
    
    {\displaystyle \angle ABC,}
  
 and 
  
    
      
        γ
      
    
    {\displaystyle \gamma }
  
 the measure of 
  
    
      
        ∠
        B
        C
        A
        .
      
    
    {\displaystyle \angle BCA.}