the city council is planning to construct a park on north street that has a triangular perimeter. they want to place a fountain at a point equidistant from all three sides of the park. where should the council place the fountain?
a. at the point of intersection of the angle bisectors and perpendicular bisectors of the park
b. at the center of the inscribed circle of the park
c. at the center of the circumscribed circle of the park
d. at the point of intersection of the lines perpendicular to two sides of the park
e. at the point of intersection of the medians of the park

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# Brief Explanations:
To find a point equidistant from all three sides of a triangle (the park), we use the property of angle bisectors. The incenter, which is the intersection of angle bisectors, is equidistant from all sides. The inscribed circle's center (incenter) is at the intersection of angle bisectors. Let's analyze each option:
- Option A: The intersection of angle bisectors (and it's also the incenter, equidistant from sides) matches the requirement. Wait, no—wait, the incenter is the intersection of angle bisectors, and it's equidistant from all sides. Wait, but let's check the options again. Wait, the problem says "equidistant from all three sides". The incenter (center of inscribed circle) is equidistant from all sides. Wait, option B says "at the center of the inscribed circle of the park"—but the center of the inscribed circle is the incenter, which is the intersection of angle bisectors. Wait, no, let's recall: the incenter is the intersection of angle bisectors, and it's equidistant from all sides. The circumcenter (center of circumscribed circle) is equidistant from all vertices, not sides. The perpendicular bisectors intersect at circumcenter (equidistant from vertices). Medians intersect at centroid (center of mass). Perpendicular to sides (altitudes) intersect at orthocenter. So the point equidistant from all sides is the incenter, which is the intersection of angle bisectors. Wait, option A: "at the point of intersection of the angle bisectors and perpendicular bisectors"—no, angle bisectors and perpendicular bisectors are different. Wait, no—wait, the incenter is only the intersection of angle bisectors. Wait, maybe I misread. Wait, the problem: "place a fountain at a point equidistant from all three sides of the park". The set of points equidistant from two sides of a triangle is the angle bisector of the angle between them. So the intersection of all three angle bisectors (incenter) is equidistant from all three sides. Now let's check the options:
- Option A: "at the point of intersection of the angle bisectors and perpendicular bisectors"—no, angle bisectors and perpendicular bisectors intersect at different points (unless it's an isoceles triangle, but generally not).
- Option B: "at the center of the inscribed circle of the park"—the center of the inscribed circle (incenter) is the intersection of angle bisectors, and is equidistant from all sides. Wait, but let's check the other options.
- Option C: "at the center of the circumscribed circle"—circumcenter is equidistant from vertices, not sides.
- Option D: "at the point of intersection of the lines perpendicular to two sides"—that's the circumcenter (perpendicular bisectors), equidistant from vertices.
- Option E: "at the point of intersection of the medians"—centroid, center of mass, not equidistant from sides.

Wait, maybe I made a mistake. Wait, the incenter is the center of the inscribed circle, and it's the intersection of angle bisectors. So the point equidistant from all three sides is the incenter, which is the center of the inscribed circle (option B) or the intersection of angle bisectors. Wait, but option A says "angle bisectors and perpendicular bisectors"—no, that's not correct. Wait, maybe the options are miswritten? Wait, no—wait, let's re-express:

- Incenter: intersection of angle bisectors, equidistant from all sides (center of incircle).
- Circumcenter: intersection of perpendicular bisectors, equidistant from all vertices (center of circumcircle).
- Centroid: intersection of medians.
- Orthocenter: intersection of altitudes (perpendicular to sides, from opposite vertex).

So the point equidistant from all sides is the incenter (center of incircle), which is option B? Wait, but option A: "angle bisectors and perpendicular bisectors"—no, angle bisectors and perpendicular bisectors are different. Wait, maybe the problem's option A is a typo, but according to the options, let's check again. Wait, the user's options:

A. at the point of intersection of the angle bisectors and perpendicular bisectors of the park

B. at the center of the inscribed circle of the park

C. at the center of the circumscribed circle of the park

D. at the point of intersection of the lines perpendicular to two sides of the park

E. at the point of intersection of the medians of the park

So the incenter is the center of the inscribed circle (option B) and is the intersection of angle bisectors. So the correct answer is B? Wait, no—wait, the incenter is the intersection of angle bisectors, and it's the center of the inscribed circle. So the point equidistant from all sides is the incenter, which is option B. Wait, but let's confirm: the distance from the incenter to each side is the radius of the incircle, so it's equidistant from all sides. So option B is correct.

Wait, but earlier I thought option A was angle bisectors, but option A is angle bisectors and perpendicular bisectors. So no, option B is the center of the inscribed circle, which is the incenter, equidistant from all sides. So the correct answer is B.

# Answer:
B. at the center of the inscribed circle of the park

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