How Ceva , s theorem and Menelaus , s theorem related?
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Theorem of Menelaus
Let three points X, Y, and Z, lie respectively on the sides AC, BC, and AB of triangle ABC. Then the points are collinear if and only if
Note that these distances are signed, so if Z lies beyond B, the ratio AZ/ZB will be negative because ZB goes in the opposite direction from AZ.
Ceva's Theorem
Three Cevians AY, BX, and CZ are concurrent at a point P if and only if
A cevian is a segment from a vertex of a triangle to any point on the side opposite except another vertex. Note that this point need not be interior to the triangle.
For Ceva's Theorem, we can use either signed distances or not; the result is the same since there will always be an even number of negative distances (zero if P is inside the triangle, two if it is outside).
So These two theorems are very useful in plane geometry because we often use them to prove that a certain number of points lie on a straight line and a certain number of lines intersect at a single point. Both of the theorems will be proved based on a common simple principle. We also generalize the theorems for arbitrary polygons
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