Measure the distance between the two end points, and divide the result by 2. This distance from either end is the midpoint of that line. Alternatively, add the two x coordinates of the endpoints and divide by 2.

The “included angle” is the angle formed by the two sides of the triangle mentioned in this theorem. This theorem is called the “Hinge Theorem” because it acts on the principle of the two sides described in the triangle as being “hinged” at their common vertex. (May also be referred to as the SSS Inequality Theorem.)

To prove the Hinge Theorem, we need to show that one line segment is larger than another. Both lines are also sides in a triangle. This guides us to use one of the triangle inequalities which provide a relationship between sides of a triangle. One of these is the converse of the scalene triangle Inequality.

The hinge theorem states that if two triangles have two congruent sides (sides of equal length), then the triangle with the larger angle between those sides will have a longer third side. The wider you open the door, the greater the hinge angle and the greater the opening length.

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

There are two important properties of midsegments that combine to make the Midsegment Theorem. The Midsegment Theorem states that the midsegment connecting the midpoints of two sides of a triangle is parallel to the third side of the triangle, and the length of this midsegment is half the length of the third side.

A midsegment is the line segment connecting the midpoints of two sides of a triangle. Since a triangle has three sides, each triangle has three midsegments. A triangle midsegment is parallel to the third side of the triangle and is half of the length of the third side.

The circumcenter of a triangle is defined as the point where the perpendicular bisectors of the sides of that particular triangle intersect. In other words, the point of concurrency of the bisector of the sides of a triangle is called the circumcenter.

One of several centers the triangle can have, the circumcenter is the point where the perpendicular bisectors of a triangle intersect. The circumcenter is also the center of the triangle’s circumcircle – the circle that passes through all three of the triangle’s vertices.

The circumcenter is equidistant from the three vertices of the triangle.

The orthocenter (H) of a triangle is the point of intersection of the three altitudes of the triangle. The circumcenter (C) of a triangle is the point of intersection of the three perpendicular bisectors of the triangle.

A centroid is the central point of a figure and is also called the geometric center. It is the point that matches to the center of gravity of a particular shape. It is the point which corresponds to the mean position of all the points in a figure. For instance, the centroid of a circle and a rectangle is at the middle.

centroid

For an acute angle triangle, the orthocenter lies inside the triangle. For the obtuse angle triangle, the orthocenter lies outside the triangle. For a right triangle, the orthocenter lies on the vertex of the right angle.

From Google:-Incenter-Circumcenter-Centroid-Orthocenter. All Of My Children Are Bringing In Peanut Butter Cookies.

: the single point in which the three bisectors of the interior angles of a triangle intersect and which is the center of the inscribed circle.

The incenter of a triangle is the point of intersection of all the three interior angle bisectors of the triangle. The angle bisectors in a triangle are always concurrent and the point of intersection is known as the incenter of the triangle.

The four ancient centers are the triangle centroid, incenter, circumcenter, and orthocenter.