Question

select all conditions that would be enough to prove that p is the circumcenter △hjk

Explanation:

Step1: Recall the definition of circumcenter

The circumcenter of a triangle is the point of intersection of the perpendicular bisectors of the sides of the triangle. Also, the circumcenter is equidistant from the vertices of the triangle.

Step2: Analyze each option

• Option 1: \(L\), \(M\) and \(N\) are midpoints of \(HK\), \(HJ\), and \(KJ\)

If \(L\), \(M\), and \(N\) are midpoints and \(PL\perp HK\), \(PM\perp HJ\), \(PN\perp KJ\), then \(P\) is the circum - center (since the circumcenter is the intersection of the perpendicular bisectors of the sides of the triangle).

• Option 2: \(PL\cong PM\cong PN\)

This only tells us that the distances from \(P\) to the sides (not the vertices) are equal. The in - center (center of the inscribed circle) is equidistant from the sides of the triangle. So this does not prove \(P\) is the circumcenter.

• Option 3: \(PK\cong HP\cong PJ\)

Since the circumcenter is equidistant from the vertices of the triangle. If \(PK = HP=PJ\), then \(P\) is the circumcenter.

• Option 4: \(KP\), \(HP\) and \(JP\) are angle bisectors of the triangle

The in - center (center of the inscribed circle) is the intersection of the angle bisectors of the triangle. So this does not prove \(P\) is the circumcenter.

Answer:

A. \(L\), \(M\) and \(N\) are midpoints of \(HK\), \(HJ\), and \(KJ\); C. \(PK\cong HP\cong PJ\)