ABC are the vertices what is he is prove that it is a triangle

Answer: Three points $A$ , $B$ , and $C$ form a triangle if and only if they are non-collinear, which can be proved by showing that $\text{Area}(\triangle ABC) \neq 0$ , or $\text{Slope}(AB) \neq \text{Slope}(BC)$ , or that the Triangle Inequality holds ( $AB + BC > AC$ ).

Step-by-step solution

Step 1: State the Non-Collinearity Condition

Three distinct points $A$ , $B$ , and $C$ in a plane form a triangle if and only if they do not all lie on the same straight line. If the points are collinear, they collapse into a single line segment and cannot enclose an area.

Step 2: Method 1: Area of Triangle Criterion

In coordinate geometry, the area enclosed by vertices $(x_1, y_1)$ , $(x_2, y_2)$ , and $(x_3, y_3)$ is given by the coordinate area formula. If this area is strictly non-zero, the three points form a valid triangle.

Step 3: Method 2: Slopes Comparison

Alternatively, compute the slopes of line segments $AB$ and $BC$ . If the slope of $AB$ is not equal to the slope of $BC$ , the lines have different directions, confirming that the three points do not lie on a common straight line.

Step 4: Method 3: Triangle Inequality Theorem

Using the distance formula, calculate the lengths of the 3 sides: $AB$ , $BC$ , and $AC$ . By the Triangle Inequality Theorem, three points form a non-degenerate triangle if the sum of the lengths of any 2 sides is strictly greater than the length of the third side.

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