Proving Ceva's Theorem with Triangle PQR

In summary, the conversation discusses the concept of tangents to an inscribed circle of a triangle and the relationship between parallel lines and concurrent lines. The relevant formula, Ceva's theorem, is also mentioned as a way to prove the concurrency of [PA], [QB], and [RC]. The use of geometry sketch pad and some additional thoughts are also mentioned.
  • #1
tomz
35
0
I have tried but still cannot get it. Simple geometry question.

Tangents to the inscribed circle of triangle PQR are parallel to [QR], [RP] and [PQ]
respectively and they touch the circle at A, B and C.
Prove that [PA], [QB] and [RC] are concurrent

relevant formula:

Ceva's theorm (Any three concurrent lines drawn from the vertices of a triangle divide the sides (produced if necessary) so that the product of their respective ratios is unity/

Thank you in advance!

I have tried on geometry sketch pad. It did works...
 
Last edited:
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  • #2
some thoughts: The parallel lines form a triangle congruent to the first rotated 180 degrees. The tangents of both triangles have a corresponding point 180 degrees apart on the circle.
 

Related to Proving Ceva's Theorem with Triangle PQR

What is Ceva's Theorem?

Ceva's Theorem is a fundamental theorem in geometry that relates the ratios of segments in a triangle to each other. It states that if three lines, each passing through a vertex of a triangle and intersecting the opposite side, are concurrent (or meet at a single point), then the ratios of the segments formed on the three sides of the triangle are equal.

How do you prove Ceva's Theorem with Triangle PQR?

To prove Ceva's Theorem with Triangle PQR, we need to show that the three lines passing through each vertex (say, PA, QB, and RC) are concurrent. This can be done by using the properties of triangles, such as the Triangle Angle Bisector Theorem, and by using basic algebraic and geometric principles.

Why is Ceva's Theorem important?

Ceva's Theorem is important because it provides a powerful tool for solving geometric problems involving ratios and proportions. It is also used in various other theorems and proofs in geometry and has applications in fields such as engineering and architecture.

What are some real-life applications of Ceva's Theorem?

Ceva's Theorem has many practical applications in fields such as surveying, navigation, and engineering. For example, it can be used to determine the optimal location for a bridge support or the ideal placement of a cell phone tower. It is also used in computer graphics to create accurate 3D models of objects and scenes.

Are there any variations of Ceva's Theorem?

Yes, there are several variations of Ceva's Theorem, including the Trigonometric Ceva's Theorem, the Menelaus Theorem, and the Stewart's Theorem. Each variation has its own set of conditions and applications, but they are all based on the same fundamental idea of ratios and proportionality in a triangle.

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