Can anyone give me the low down on Ceva's Theorem?

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In summary, Ceva's theorem is a geometric theorem that states that in a triangle, the three lines drawn from each vertex to the opposite side, are concurrent if and only if the product of the ratios of the segments on each side is equal to 1.
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tamintl
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Hi all,

Can anyone give a quick few lines on what Ceva's theorem consists of?


For example,

'Let ABC be a traingle and suppose there is a circle inside the triangle which is tangent to all three sides. Let it be tangent to BC at P and tangent to CA at Q, and tangent to AB at R.
--> How could i use Ceva's theorem to show that AP, BQ, and CR are concurrent?'

Im not looking for a proof. Just the steps i should take. Any help would be appreciated!

Regards
Tam
 
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  • #2
One needs to calculate the ratios BQ/QC &c. & show that the product is 1. The angular version of Ceva's theorem is often easier to use.
 

Related to Can anyone give me the low down on Ceva's Theorem?

1. What is Ceva's Theorem?

Ceva's Theorem is a mathematical theorem that states that in a triangle, if three lines are drawn from each vertex to the opposite side, then the three lines are concurrent (meet at a single point) if and only if the product of the ratios of the lengths of the three segments formed by the point of concurrency on each side is equal to 1.

2. Who is Ceva and how did this theorem come about?

Giovanni Ceva was an Italian mathematician who first published the theorem in 1678 in his book "De linearum situ" (On the location of lines). However, the theorem was known to ancient Greek mathematicians, particularly Thales, who is said to have used it to find the height of the Great Pyramid of Giza.

3. What is the significance of Ceva's Theorem?

Ceva's Theorem is significant because it provides a concise and elegant way to prove the concurrency of lines in a triangle. It is also used in various fields such as geometry, trigonometry, and physics to solve problems and make calculations.

4. How is Ceva's Theorem related to other theorems in geometry?

Ceva's Theorem is closely related to other theorems in geometry, such as Menelaus' Theorem and Stewart's Theorem. These theorems all deal with the relationships between the sides and angles of a triangle and can be used to solve various problems involving triangles.

5. Can you provide an example of how to use Ceva's Theorem?

Sure! Let's say we have a triangle ABC with sides AB, BC, and CA, and we draw three lines from each vertex to the opposite side, intersecting at a point P. According to Ceva's Theorem, if AP, BP, and CP are concurrent, then (BD/DC) x (CE/EA) x (AF/FB) = 1, where D, E, and F are the points of intersection of AP, BP, and CP with the opposite sides respectively. We can use this ratio to find the length of a side or the measure of an angle in the triangle.

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