Are perpendicular bisectors always altitudes? Are altitudes always perpendicular bisectors? Does the circumcenter and the incenter occur at the same place? Exceptions?

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## Answers (2)

No, perpendicular bisector are not always altitudes.

Perpendicular bisector is a line that pass through a segment's midpoint making 90 degree angles.

Altitude of a triangle is the segment from the vertice to the opposite side so that the segment is perpendicular to the side.

Picture a scalene triangle with perpendicular bisectors, you'll see that perpendicular bisectors are not always altitudes. They could be, such as in an equilateral triangle.

No, altitudes are not always perpendicular bisectors. Because the side parts that the altitude divide does not necessarily congruent. They could be the same, such as in case of an equilateral triangle.

No, the circumcenter and the incenter does not always occur at the same place. They could be, such as in case of an equilateral triangle.

EXPLANATIONS:

The circumcenter of a triangle is equidistant from the three vertices, and so it is also the radius of a circle that passes through the vertices.

Can be construct by drawing perpendicular bisectors of the sides of the triangle, the point 3 lines intersect is circumcenter.

Mnemonic aide: "circum" means around, so the circumcenter of a triangle is the center of the circle that goes around the triangle.

Visual here:

The incenter is equidistant from the three sidelines, and so it is also the radius of a circle that is tangent to the sidelines (in other words, in the triangle).

Can be construct by drawing angle bisectors of the triangle, the point 3 lines intersect is the incenter.

Mnemonic aide: "in" means, well, in; so the incenter of a triangle is the center of the circle that is in that triangle.

Visual here:

In a right triangle the circumcenter is the midpoint of the hypotenuse, but the incenter is inside the triangle, so they cannot be the same point.