ত্রিভুজের গুরূত্ব

The triangle is important for many reasons.

First, understanding triangles helps understand all other polygons because they can be broken down into triangles. They are the second simplest thing to lines, therefore the simplest closed figures (arguably circles, too). One interesting corollary of this fact is that the sum of angles in a triangle in the plane always adds up to $180^\circ$, which is a critical advantage that lines don't have.

Second, we have many simple concurrencies and collinearities. We have all of the crazy triangle centers, the nine-point circle, the Simson lines, we have Menelaus' Theorem, we have Ceva's Theorem. The closest thing we can come to defining a center of a higher-order polygon is the center of mass, but we have many options in triangles.

Third, triangles always have a unique circumcircle and a unique incircle. This is very helpful in many cases, especially in constructions.

Fourth, triangles make up unique planes, so they're extremely helpful in 3D geometry.

There are many more interesting properties that make triangles essential in our understanding to geometry.

Before I took the final exam for my freshman geometry course, I put together a study guide covering the basics of what we learned during the year. I titled it "Honors Triangles: The Essentials". Why?

From All, Triangles

Excluding the most basic notions of point, line, and angle, the triangle can be considered the most important object in all of basic geometry. All polygons can be broken up into different triangles.

Circles? With some help from Archimedes, we can show that it's possible to get a really good approximation for the area of a circle using cleverly arranged narrow triangles, with a perfect approximation possible if we allow one of the sides of each triangle to be infinitesimally small.

From Triangles, All

Just as many important parts in geometry are composed of triangles, limitless possibilities regarding manipulation, synthesis, and calculation lie within the triangle itself. There are similar triangles, congruent triangles, a plethora of different centers of a triangle...