If the point inside a triangle are equidistant from the vertex of the same triangle, what the name of that point is ?

Options

A. Incentre
B. Orthocentre
C. Median
D. Circumcentre

Answer

To solve the question, we need to identify the point inside a triangle that is equidistant from all three vertices of the triangle. Here's a step-by-step solution:

### Step 1: Understand the Triangle and Points
We have a triangle, which we can label as triangle ABC. The vertices of the triangle are A, B, and C.

**Hint:** Visualize the triangle and label the vertices clearly.

### Step 2: Define the Point Inside the Triangle
Let’s denote the point inside the triangle as point O. According to the problem, the distances from point O to each vertex of the triangle are equal. This means that OA = OB = OC.

**Hint:** Remember that the distances from point O to each vertex are crucial to identifying the type of point.

### Step 3: Identify the Characteristics of Point O
Since point O is equidistant from all three vertices of triangle ABC, it satisfies a specific condition that is characteristic of a well-known point in triangle geometry.

**Hint:** Think about the special points in triangle geometry that relate to distances from vertices.

### Step 4: Determine the Name of the Point
The point O, which is equidistant from the vertices A, B, and C, is known as the **Circumcenter** of the triangle. The circumcenter is the point where the perpendicular bisectors of the sides of the triangle intersect, and it is equidistant from all three vertices.

**Hint:** Recall the definitions of various triangle centers (like centroid, orthocenter, and circumcenter) to confirm your answer.

### Conclusion
The point inside the triangle that is equidistant from the vertices is called the **Circumcenter**.

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