ABC is a triangle, if perpendicular distance from particular point in same plane to 3 sides of a triangle .How many such points are exist?

Question

Answer

Only one point

Answer

There is only one point exists

Answer

Points in a Triangle with Perpendicular Distances

Introduction:
In a triangle ABC, if we have a particular point in the same plane and draw perpendicular distances from this point to all three sides of the triangle, we can determine how many such points exist.

Key Points:
- For any point inside the triangle, the perpendicular distances to all three sides will be less than the altitudes from the vertices.
- For any point outside the triangle, the perpendicular distance to one of the sides will be greater than the altitude from the opposite vertex.
- Therefore, there exists only one point in the plane of the triangle that has equal perpendicular distances to all three sides. This point is called the incenter of the triangle.

Conclusion:
In conclusion, there exists only one point in the same plane as a triangle that has equal perpendicular distances to all three sides, and this point is known as the incenter of the triangle. This point plays a significant role in various geometric constructions and theorems related to triangles.

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