Talk:Section formula

Coordinates of centroid
The centroid of a triangle is the intersection of the medians and divides each median in the ratio $2:1$. Let the vertices of the triangle be $$A(x_1, y_1)$$, $B(x_2, y_2)$ and $C(x_3, y_3)$. So, a median from point A will intersect BC at $\left(\frac{x_2 + x_3}{2}, \frac{y_2 + y_3}{2}\right)$. Using the section formula, the centroid becomes: $$ G = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right)

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Coordinates of incenter
Let the sides of a triangle be $a$, $b$ and $c$  its vertices are $A(x_1, y_1)$ , $B(x_2, y_2)$  and $C(x_3, y_3)$. The Incentre (intersection of the angle bisectors) divides the angle bisectors in the ratio $(b+c):a$, $(a+c):b$ and $(a+b):c$. An angle bisector also divides the opposite side in the ratio of the adjacent sides (Angle bisector theorem). So they meet at $\left(\frac{bx_2+cx_3}{b+c}, \frac{by_2+cy_3}{b+c}\right)$. Thus, the incenter is $$ I = \left(\frac{ax_1 + bx_2 + cx_3}{a+b+c}, \frac{ay_1 + by_2 + cy_3}{a+b+c} \right) $$ This is essentially the weighted average of the vertices.

Shubhrajit Sadhukhan (talk) 13:25, 7 November 2020 (UTC)