MathBin	Triangle	Member

A triangle is a two-dimensional object that exists in two dimensions or higher. A triangle is a polygon with three edges (sides) and three vertices (angles), and is one of the basic shapes in geometry. In Euclidean geometry, any three points that are non-collinear determine a unique triangle and a unique plane.

The three angles in a triangle are labeled A, B, and C, and the three sides across from each angle are labeled a, b, and c. So side a is across from angle A, side b is across from angle B, and side c is across from angle C. If you know any three of these six measurements (aka facts), you can (in most cases) construct a triangle using either the Law of Cosines or the Law of Sines.

The angles within a triangle must add up to 180 degrees (A + B + C = 180°). A triangle with all angles less than 90 degrees is an Acute triangle. A triangle with one 90 degree angle is a Right triangle. And a triangle with one angle more than 90 degrees is an Obtuse triangle.

An Equilateral triangle has three equal sides, and all three angles are 60 degrees. An Isosceles triangle has two equal sides with two equal angles opposite them. A Scalene triangle has no equal sides, and no equal angles.

The first three bins given below show how to construct a triangle with the given information. Subsequent sections will describe each of these constructors.

1. Finds the triangle given three of its six angles and sides.
2. Finds the triangle given by its vertices.
3. Finds the triangle given by the end points of three vectors.
4. Finds the triangle given by three points on its circumscribed circle.
5. Finds the triangle with maximum area given by three points on its circumscribed circle.

A triangle is classified based on what three of the six measurements are given. Note that A means angle and S means side.

• AAA - Given all three angles, there is no unique triangle.
• AAS - Given any two angles and one side, use the Law of Sines to construct the triangle.
• SAS - Given two sides and the angle between them, use the Law of Cosines, then the Law of Sines to construct the triangle.
• SSA - Given two sides and a non-included angle, use the Law of Sines to construct a triangle(s). There might be two possible triangles.
• SSS - Given all three sides, use the Law of Cosines to construct the triangle.

The law of cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. If the angle is a right angle, then the law of cosines reduces to the Pythagorean theorem a² + b² = c². The law of cosines finds the third side of a triangle when two sides and their enclosed angle are known (SAS), and in computing the angles of a triangle if all three sides are known (SSS).

• a² = b² + c² - 2 b c cos A
• b² = a² + c² - 2 a c cos B
• c² = a² + b² - 2 a b cos C

The law of sines relates the lengths of the sides of a triangle to the sines of its angles.

(sin A)/a = (sin B)/b = (sin C)/c

Since two angles are given, and all three angles of a triangle must add up to 180°, the third angle is easy to find. Then use the law of sines to find the other two sides.

1. Find angle 3: A₃ = 180° - A₁ - A₂
2. Find side 2: S₂ = S₁ * sin(A₂) / sin(A₁)
3. Find side 3: S₃ = S₁ * sin(A₃) / sin(A₁)

Since two sides and the angle between them are given, use the law of cosines to find the third side. Then use the law of sines to find the smaller of the two angles based on their opposite sides. This is because the sine of an angle cannot tell if the angle is acute or obtuse. The third angle plus the other two must add up to 180°. The following steps assume the second angle is acute.

1. Find side 3: S₃ = S₁² + S₂² - 2S₁S₂ cos(A₃)
2. Find angle 2: A₂ = arcsin(S₂ sin(A₃) / S₃)
3. Find angle 3: A₃ = 180° - A₁ - A₂

Since two sides and the angle not between them are given, use the law of sines to find the unknown angle across from one of the sides you are given. Say we know sides S₁, S₂, and angle A₁. If S₁ < S₂ and 0 < (S₂ / S₁) sin(A₁) < 1, then there are two values for angle A₂ where one is acute and the other is obtuse, and these two angles are supplementary. The third angle is easy to find once you have the other angles. You can find the third side using either the law of sines or the law of cosines (I used the law of sines).

1. Find angle 2: A₂ = arcsin(S₂ sin(A₁) / S₁. Check if there are two possible values.
2. Find angle 3: A₃ = 180° - A₁ - A₂
3. Find side 3: S₃ = S₁ sin(A₃) / sin(A₃)

1. A = 25°, a = 5, b = 11 → B is 68.39746° or 111.60254°, c is 11.81021, or c = 8.12856.

Since no angles are given, the law of cosines must be used to find the first angle. Then you can use either law to find the second angle. And the third angle is 180° minus the other two angles. The steps below use the law of cosines to find each angle.

1. Find angle 1: A₁ = cos^-1[(S₂² + S₃² - S₁²) / (2S₂S₃)]
2. Find angle 2: A₂ = cos^-1[(S₃² + S₁² - S₂²) / (2S₃S₁)]
3. Find angle 3: A₃ = cos^-1[(S₁² + S₂² - S₃²) / (2S₁S₂)]

This bin allows you to pick any three of the six measurements for a triangle, and determines if either one, two, or no triangles are possible. The following lists the 16 possible combinations you can select.

1. A, B, C (AAA) - No triangle is possible.
2. A, B, a (AAS) - Law of Sines
3. A, B, b (AAS) - Law of Sines
4. A, B, c (ASA) - Law of Sines
5. A, C, a (AAS) - Law of Sines
6. A, C, b (ASA) - Law of Sines
7. A, C, c (AAS) - Law of Sines
8. B, C, a (ASA) - Law of Sines
9. B, C, b (AAS) - Law of Sines
10. B, C, c (AAS) - Law of Sines
11. A, a, b (SSA) - Law of Sines. Two triangles are possible.
12. A, a, c (SSA) - Law of Sines. Two triangles are possible.
13. A, b, c (SAS) - Law of Cosines
14. B, a, b (SSA) - Law of Sines. Two triangles are possible.
15. B, a, c (SAS) - Law of Cosines
16. B, b, c (SSA) - Law of Sines. Two triangles are possible.
17. C, a, b (SAS) - Law of Cosines
18. C, a, c (SSA) - Law of Sines. Two triangles are possible.
19. C, b, c (SSA) - Law of Sines. Two triangles are possible.
20. a, b, c (SSS) - Law of Cosines