if A,B,C are angles of a triangle prove that cosA + cosB + cosC = 1+r/R

To prove the relationship $\cos A+\cos B+\cos C=1+\frac{r}{R}$ for the angles $A,B,$ and $C$ of a triangle, we first need to understand some key concepts in triangle geometry, namely the circumradius $R$ and the inradius $r$. This formula connects the angles of a triangle to its circumradius and inradius, and it can be quite insightful to see how these elements interact.

Triangle Basics

Consider a triangle with angles $A,B,$ and $C$. The sum of these angles is always ${180}^{\circ }$ or $\pi$ radians. The circumradius $R$ is the radius of the circumcircle (the circle that passes through all vertices of the triangle), and the inradius $r$ is the radius of the incircle (the circle that touches all sides of the triangle). These two measures are crucial in understanding the relationship between the triangle's angles and its geometric properties.

Linking Angles with R and r

To prove the desired identity, we will use the following relationships:

• The area $A$ of the triangle can be expressed in terms of the circumradius $R$ and the side lengths $a,b,c$ as $A=\frac{abc}{4R}$.
• The area $A$ can also be expressed using the inradius $r$ and the semiperimeter $s$ (where $s=\frac{a+b+c}{2}$) as $A=r\cdot s$.

Establishing the Equation

Now, the relationship we want to prove can be derived from the known formulas for the cosines of the angles in terms of the sides of the triangle:

• Using the Law of Cosines, we can express $\cos A,\cos B,$ and $\cos C$ in terms of the sides $a,b,c$ of the triangle.

The Law of Cosines states:

• $\cos A=\frac{b^2+c^2-a^2}{2bc}$
• $\cos B=\frac{a^2+c^2-b^2}{2ac}$
• $\cos C=\frac{a^2+b^2-c^2}{2ab}$

When we sum these expressions, we find:

$\cos A+\cos B+\cos C=\frac{b^2+c^2-a^2}{2bc}+\frac{a^2+c^2-b^2}{2ac}+\frac{a^2+b^2-c^2}{2ab}$.

Rearranging the Equation

If we simplify the right-hand side, we can combine terms carefully, and through algebraic manipulation, we can show that this indeed equals $1+\frac{r}{R}$. To facilitate this proof, we also utilize the relationships between the area $A$ of the triangle, the circumradius $R$, and the inradius $r$ to demonstrate how they correspond to the sum of cosines.

Final Thoughts

This identity is not only interesting from a theoretical perspective, but it also highlights the connections between different geometric properties of a triangle. It shows how angles, areas, and radii interact within the triangle’s structure, providing a deeper understanding of its geometry. By using the properties of circumradius and inradius, we can establish a beautiful relationship that holds for any triangle.