Question

Point Z is the circumcenter of ΔLMN.

Point Z is the circumcenter of triangle L M N. Lines are drawn from each point of the triangle to point Z. Lines are drawn from point Z to each side to form right angles and line segments Z A, Z B, and Z C.
Which must be true?

Line segment A N is-congruent-to line segment L B
Line segment N C is-congruent-to line segment B Z
AngleBMZ Is-congruent-to AngleCMZ
AngleZBM Is-congruent-to AngleZCM

Answer 1

None of them must be true.

Reason: If Z is the circumcenter then ZA, ZB, ZC are perpendiculars to the sides, so A, B, C are the midpoints of the sides. That makes

- AN = 1/2·MN while LB = 1/2·LN, so AN = LB only when MN = LN (not always).
- BZ is the distance from the circumcenter to a side, while NC is the distance from a vertex to the midpoint of the opposite side — these need not be equal.
- Angle BMZ and angle CMZ are angles between MZ and two different lines (MB and MC); they are equal only in special triangles, not in general.
- Angle ZCM = 90° (ZC ⟂ LM) but angle ZBM need not be 90°, so those angles are not generally congruent.

Concrete counterexample: take L=(0,0), M=(4,0), N=(1,3). Then Z=(2,1), A=(2.5,1.5), B=(0.5,1.5), C=(2,0). Numerical checks show AN ≠ LB, NC ≠ BZ, ∠BMZ ≠ ∠CMZ, and ∠ZBM ≠ ∠ZCM. Thus none of the four statements must hold for every triangle.