A, B C and D are four points in a plane with position vectors, `veca, vecb vecc and vecd` respectively, such that `(veca-vecd).(vecb-vecc)= (vecb-vecd).(vecc-veca)=0` then point D is the ______ of triangle ABC.

Explanation

To solve the problem, we need to analyze the given conditions step by step.

### Step 1: Understand the Given Conditions
We have four points A, B, C, and D with position vectors \( \vec{a}, \vec{b}, \vec{c}, \vec{d} \) respectively. The conditions given are:
1. \( (\vec{a} - \vec{d}) \cdot (\vec{b} - \vec{c}) = 0 \)
2. \( (\vec{b} - \vec{d}) \cdot (\vec{c} - \vec{a}) = 0 \)

### Step 2: Interpret the Dot Product Conditions
The dot product being zero indicates that the vectors are perpendicular. Therefore:
- From the first condition, \( \vec{a} - \vec{d} \) is perpendicular to \( \vec{b} - \vec{c} \).
- From the second condition, \( \vec{b} - \vec{d} \) is perpendicular to \( \vec{c} - \vec{a} \).

### Step 3: Analyze the Geometric Implications
1. **Perpendicularity of Vectors**:
- Since \( \vec{a} - \vec{d} \) is perpendicular to \( \vec{b} - \vec{c} \), it means that the line segment AD is perpendicular to the line segment BC.
- Similarly, since \( \vec{b} - \vec{d} \) is perpendicular to \( \vec{c} - \vec{a} \), the line segment BD is perpendicular to the line segment CA.

### Step 4: Conclusion about Point D
The point D is positioned such that:
- The line AD is perpendicular to BC.
- The line BD is perpendicular to CA.

This configuration indicates that point D is the orthocenter of triangle ABC. The orthocenter is the point where the altitudes of a triangle intersect.

### Final Answer
Thus, point D is the **orthocenter** of triangle ABC.

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