Question

what is the angle?
key ideas:
orthocenter: the three ____________ of a triangle meet at the orthocenter.
circumcenter: the perpendicular __________ of each side of a triangle meet at the circumcenter. the circumcenter is __________ from each of the vertices of the triangle.
incenter: the angle __________ of each angle of a triangle meet at the incenter. the incenter is __________ from each of the sides of the triangle.
centroid: the __________ of a triangle connects from a vertex on the triangle to the __________ of the side opposite the vertex.
the ____________ meet at the centroid.
the centroid is ____________ of the distance from each vertex to the midpoint of the opposite side.

Explanation:

• Orthocenter: By definition, the three altitudes of a triangle meet at the orthocenter.
• Circumcenter: The perpendicular bisectors of each side of a triangle meet at the circumcenter. And the circumcenter is equidistant from each of the vertices of the triangle (since it is the center of the circum - circle passing through all three vertices).
• Incenter: The angle bisectors of each angle of a triangle meet at the incenter. The incenter is equidistant from each of the sides of the triangle (as it is the center of the incircle tangent to all three sides).
• Centroid: The median of a triangle connects from a vertex on the triangle to the mid - point of the side opposite the vertex. The medians meet at the centroid. The centroid is two - thirds of the distance from each vertex to the midpoint of the opposite side (a well - known property of the centroid in a triangle).

Answer:

• Orthocenter: altitudes
• Circumcenter: bisectors; equidistant
• Incenter: bisectors; equidistant
• Centroid: median; mid - point; medians; two - thirds