Question

H is the circumcenter, or point of concurrency, of the perpendicular bisectors of ΔACE.

Triangle A C E is shown. Point H is the circumcenter of the triangle. Lines are drawn from each point of the triangle to point H. Lines are drawn from point H to the sides of the triangle to form right angles. Line segments H B, H F, and H D are formed.
Which statements must be true regarding the diagram?

∠HDC ≅ ∠HDE
AB ≅ AF
AB ≅ BC
HC ≅ HE
HA ≅ HB

Answer 1

True:
- ∠HDC ≅ ∠HDE — D is the foot of the perpendicular from H to CE, so HD ⟂ CE; both angles are right angles.
- HC ≅ HE — H is the circumcenter, so H is equidistant from the vertices: HA = HC = HE.

Not necessarily true:
- AB ≅ AF — B and F are different feet on different sides; those segment lengths need not be equal.
- AB ≅ BC — no reason this must hold.
- HA ≅ HB — HA is a radius (distance to vertex A) while HB is the distance from H to side AE; they are not generally equal.