Animated Solution for Mathematics - Straight Lines: The incentre of the triangle with vertices $(1,\sqrt{3}),(0,0)$ and $(2,0)$ is

Welcome, future engineer. Today, we are going to solve a problem that seems like a standard coordinate geometry exercise but is actually a profound lesson in observation.

We are given three vertices: $A(1,\sqrt{3})$, $B(0,0)$, and $C(2,0)$. When you see these coordinates, don't just jump into the formula. Stop, breathe, and visualize.

Plotting these points on the Cartesian plane reveals a beautiful, symmetric structure. Point $B$ is at the origin, and $C$ sits on the $x$-axis. Point $A$ is perched above, creating a triangle.

Now, let's calculate the side lengths. Using the distance formula $d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$, we find the lengths of the sides:

This is the 'Aha!' moment. All sides are equal, confirming that we have an equilateral triangle.

In the world of geometry, symmetry is a gift. For an equilateral triangle, the incentre, centroid, circumcentre, and orthocentre all collapse into a single point.

This means we don't need the complex incentre formula. We can simply find the centroid, which is the average of the coordinates:

Plugging in our values, we calculate the coordinates:

Our incentre is located at the point:

This problem teaches us that the most efficient path is often found by observing the properties of the figure before applying the formulas. Keep this mindset, and you will conquer the JEE.