Problem

A sprinkler must be equally far from the \(\text{three}~\text{corner}~\text{points}\) of a triangular garden. Identify the appropriate triangle center, its defining construction lines, the equal-distance relationship, its circle role, and whether it must lie inside the garden.

Source triangular garden with the three corner points identified as the equal-distance targets.

Big Picture

What this problem is really about

The decisive question is what must be equally distant: vertices or side lines. We’ll match the three corner-point distances to their defining loci, intersect two of those loci to identify the center, and then check how the triangle’s shape affects whether that point lies inside.

Answer choices

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Preview only

Use the circumcenter: intersect side perpendicular bisectors to get a point equally distant from the three vertices.

Use the circumcenter as a point equally distant from the three side lines.

Use the incenter: intersect angle bisectors to get a point equally distant from the vertices.

Use the circumcenter without constructing its defining lines.

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A sprinkler must be equally far from the \(\text{three}~\text{corner}~\text{points}\) of a triangular garden. Identify the appropriate triangle center, its defining construction lines, the equal-distance relationship, its circle role, and whether it must lie inside the garden.

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