In 1678, Giovanni Ceva (1647 - 1734) published a useful theorem.
A CEVIAN is a straight line segment from a vertex to a point on the side opposite that vertex.
The diagram shows â–³ ABC and three cevians AD, BE, CF which are drawn so as to intersect - (be CONCURRENT) - at point X
Ceva's theorem is that
The converse of Ceva's theorem is also true so that three cevians are concurrent if and only if this relationship is true.
In this way it can be proved that the medians of a triangle are concurrent. Also concurrent are the three internal angle bisectors and the three altitudes.
Proofs
The following links offer proofs of Ceva' theorem and its converse. Other proofs are available.
Art of Problem Solving - Ceva's theorem
Byjus - Ceva's theorem
Testbook.com - Ceva's theorem
Concurrencies
In a triangle, there are four main points of concurrency: the centroid, orthocentre, circumcentre, and incentre.
The centroid is the point of concurrency where the three medians of a triangle intersect. It divides each median into segments whose lengths are in a ratio of 2, with the longer portion of each median closer to each vertex. The centroid is always within the triangle and is also the triangle's center of mass or center of gravity.
Centroid - Wikipedia
The medians of a triangle join the vertices to the midpoints of the opposite sides. The concurrency of the medians is easily proved by Ceva's theorem. The proof is shown in the next diagram.
The orthocentre is the point of concurrency where the three altitudes of a triangle intersect. The orthocentre is not always found on the interior of a triangle. In an obtuse triangle, the orthocentre is found outside of the triangle. In a right triangle, it is located on the right angle. In an acute triangle, the orthocentre is located on the interior of the triangle.
Orthocentre - Wikipedia
The circumcentre is the point of concurrency of the perpendicular bisectors of the sides of the triangle. In an obtuse triangle, the circumcentre is located outside of the triangle. In a right triangle, it is located on the hypotenuse. In an acute triangle, the circumcentre is located on the interior of the triangle.
The cirumcentre is the centre of the circumcircle - that is, the circle which passes through all three vertices of the triangle.
Circumcentre of Triangle - Cuemath.com
The incentre is the point of concurrency where the angle bisectors of a triangle intersect. It is the centre of the triangle's incircle, the largest circle that can be drawn inside the triangle and is tangential to all three sides.
No comments:
Post a Comment