Proving an Angle of Triangle ABC is Greater Than 90 Degrees Under Certain Conditions

When examining a triangle ABC, we often seek a method to determine whether one of its angles is greater than 90 degrees under certain conditions. This article explores how to prove this by leveraging vector algebra and dot products.

Using Vector Algebra and Dot Products to Analyze Angles

Let's consider a triangle ABC where the sides are represented as directed line segments or bound/fixed vectors. The measures of the three angles at vertices A, B, and C can be calculated using the dot products of these vectors. Vectors AB, BC, and CA are typically denoted by small lowercase letters corresponding to the uppercase letters denoting the opposite vertices. For instance, if the vertices are A(x1, y1), B(x2, y2), and C(x3, y3), the vector AB can be represented as (x2 - x1, y2 - y1).

Steps to Calculate the Angles Using Dot Products

Step 1: Represent the three sides of the triangle as vectors: AB (x2 - x1, y2 - y1), BC (x3 - x2, y3 - y2), and CA (x1 - x3, y1 - y3). Step 2: Calculate the dot products of these vectors: AB . AC, BA . BC, and CA . CB. Step 3: Use the formula for the dot product: AB . AC c b cosα, BA . BC c a cosβ, and CA . CB b a cosγ, where α, β, and γ represent the angles at vertices A, B, and C respectively.

Note that the orientation/sense of these vectors is crucial. For example, the vector AB goes from point A to point B, whereas the vector BA goes from B to A, leading to a different coordinate representation.

Identifying an Obtuse Angle Using Angles and Sides

Another method involves examining the measures of the angles of triangle ABC. If the measures of two angles fall in the interval 0 to π/4 or their sum is exactly π/2, then the third angle must be π/2, and the triangle is obtuse-angled.

Example Conditions for an Obtuse Triangle

If α and β are both in the interval [0, π/4], or α β π/2, then the third angle γ must be greater than π/2 and the triangle is obtuse-angled.

Role of Projections in Analyzing Angles

Projections, especially vector projections and scalar projections, play a significant role in understanding angles. The dot product is an essential tool in calculating these projections. For example, the vector projection of a onto b (Projb a) and the scalar projection ab (prb a) are connected to the dot product. If any one of the dot products mentioned is zero, it indicates that the respective angle is π/2.

For a detailed explanation, refer to Alexandru Carausu (2003)'s book 'Vector Algebra, Analytic and Differential Geometry', where you can find how these concepts are applied in the context of triangles and higher dimensions.

In summary, by employing vector algebra and dot products, we can systematically analyze the angles of a triangle and determine if any of them exceed 90 degrees. This approach combines the power of vector mathematics with the properties of dot products, providing a robust method for triangle analysis.