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The concept of concurrency in triangles, specifically the concurrency of medians, altitudes, perpendicular bisectors, and angle bisectors. It explains the properties and relationships of these lines and the points of concurrency, including the centroid, circumcenter, and incenter. It also provides examples and equations to find the coordinates of these points and the lengths of medians.
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Concurrent lines are three or more lines that intersect at the same point. The mutual point of intersection is called the point of concurrency.
M is the point of concurrency of lines M w, y, and x. w x y The Four Centers of a Triangle In a triangle, the following sets of lines are concurrent: · The three medians. · The three altitudes. · The perpendicular bisectors of each of the three sides of a triangle. · The three angle bisectors of each angle in the triangle. Concurrency of the Medians The median of a triangle is the line segment that joins the vertex to the midpoint of the opposite side of the triangle. The three medians of a triangle are concurrent in a point that is called the centroid. There is a special relationship that involves the line segments when all of the three medians meet. The distance from each vertex to the centroid is two-thirds of the length of the entire median drawn from that vertex Let's Take a Look at the Diagram.... O is the centroid of Δ ABC, points D, F, and E are midpoints.
In addition, the distance from each centroid to the opposite side (midpoint) is one-third of the distance of the entire median. O is the centroid of Δ ABC, points D, F, and E are midpoints. The centroid also divides the median into two segments in the ratio 2:1, such that: and (^) and If you notice, the bigger part of the ratio is the segment that is drawn from the vertex to the centroid. The smaller part of the median is always the part that is drawn from the centroid to the midpoint of the opposite side. When working with these ratios, it is important to never mix the two up!!!
Properties of the Circumcenter The circumcenter is the center of the circle that can be circumscribed around the triangle.
The incenter is the center of a circle that is inscribed inside a triangle.
Concurrency of the Altitudes An altitude of a triangle is a line segment that is drawn from the vertex to the opposite side and is perpendicular to the side. There are three altitudes in a triangle. The altitudes of a triangle, extended if necessary, are concurrent in a point called the orthocenter of the triangle. Location of the Orthocenter The orthocenter can fall in the interior of the triangle, on the side of the triangle, or in the exterior of the triangle. Right Triangle The orthocenter is located on the right angle. Acute Triangle The orthocenter is located inside the triangle.
The orthocenter is located outside the triangle.