Triangle Centers: Orthocenter and Incenter

The orthocenter of a triangle is the point where all three altitudes of the triangle intersect.
An altitude of a triangle is a line segment from a vertex to the opposite side (or its extension) that is perpendicular to that side.
In an acute triangle, the orthocenter lies inside the triangle.
In a right triangle, the orthocenter is located at the vertex where the right angle is formed.
In an obtuse triangle, the orthocenter lies outside the triangle.
To find the orthocenter, determine the equations of two altitudes and solve for their point of intersection.
The orthocenter is one of the triangle's centers.

The incenter of a triangle is the point where the three angle bisectors of the triangle intersect.
An angle bisector is a line segment that divides an angle into two equal angles.
The incenter is always inside the triangle, regardless of whether the triangle is acute, right, or obtuse.
The incenter is the center of the triangle's incircle – the largest circle that can be inscribed inside the triangle.
The incenter is equidistant from all three sides of the triangle.
The distance from the incenter to each side of the triangle is the radius of the incircle.
To find the incenter, determine the equations of two angle bisectors, and solve for their point of intersection.
The incenter is one of the triangle's centers.

A midsegment of a triangle is a line segment that connects the midpoints of two sides of the triangle.
Every triangle has three midsegments.
A midsegment is parallel to the third side of the triangle (the side not connected to the midsegment's endpoints).
The length of a midsegment is exactly half the length of the third side to which it is parallel.
The midsegment forms a smaller triangle that is similar to the original triangle; the ratio of corresponding sides is 1:2.
The four triangles formed by the three midsegments are congruent.
The three midsegments divide the triangle into four congruent triangles, each similar to the original triangle.

The centroid of a triangle is the point where the three medians of the triangle intersect.
A median of a triangle is a line segment from a vertex to the midpoint of the opposite side.
The centroid is always inside the triangle.
The centroid divides each median in a 2:1 ratio, with the longer segment being between the vertex and the centroid.
The centroid is the center of mass of the triangle. If you were to cut out a triangle from a piece of cardboard, it would balance on a pin placed at the centroid.
To find the centroid, determine the equations of two medians, and solve for their point of intersection.
The centroid is one of the triangle's centers.

The circumcenter of a triangle is the point where the three perpendicular bisectors of the sides of the triangle intersect.
A perpendicular bisector is a line that is perpendicular to a side of the triangle and passes through its midpoint.
The circumcenter is the center of the triangle's circumcircle – the circle that passes through all three vertices of the triangle.
The circumcenter is equidistant from the three vertices of the triangle.
In an acute triangle, the circumcenter lies inside the triangle.
In a right triangle, the circumcenter lies on the midpoint of the hypotenuse.
In an obtuse triangle, the circumcenter lies outside the triangle.
To find the circumcenter, determine the equations of two perpendicular bisectors, and solve for their point of intersection.
The circumcenter is one of the triangle's centers.

A triangle has many centers, each defined by a specific property.
The incenter, centroid, orthocenter, and circumcenter are four of the most well-known triangle centers.
These centers may or may not coincide, depending on the type of triangle.
In an equilateral triangle, all four centers (incenter, centroid, orthocenter, and circumcenter) coincide at a single point.
The Euler line is a line that passes through the orthocenter, circumcenter, and centroid of any triangle that is not equilateral.
The incenter does not necessarily lie on the Euler line.
The centroid always lies between the orthocenter and the circumcenter on the Euler line, and the distance from the orthocenter to the centroid is twice the distance from the centroid to the circumcenter.
Understanding the properties and relationships between these centers is a key concept in geometry.

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