Proving PM MQ in the Butterfly Theorem
The Butterfly Theorem is a fascinating result in circle geometry that often goes by verbal descriptions. However, proving this theorem rigorously involves the use of coordinate geometry and properties of similar triangles. This article will guide you through the detailed steps to prove that PM MQ in the context of the Butterfly Theorem.
What is the Butterfly Theorem?
The Butterfly Theorem states that if through the midpoint M of chord AB of a circle two chords CD and EF are drawn, and if ED and CF intersect AB in points P and Q respectively, then PM MQ.
Proof: PM MQ
To prove that PM MQ, we will follow a structured approach based on properties of circles and similar triangles.
Establish Coordinates
- Consider a circle centered at the origin O(0, 0) with radius r. Let the coordinates of points be as follows:
A (-a, b) B (a, b) M, the midpoint of AB, has coordinates M(0, b).Draw Chords
- Let the lines CD and EF intersect AB at points P and Q respectively.
Use Perpendicularity
- The line segment OM is perpendicular to the chord AB because the radius to the midpoint of a chord is perpendicular to the chord itself. Therefore, OM ⊥ AB.
Consider Triangles
- Since M is the midpoint of AB, we can analyze triangles OMP and OMQ.
By the properties of circles, the angles ∠OMP and ∠OMQ are equal because they subtend the same arc OP and OQ on the circle.Apply Similar Triangles
- Triangles OMP and OMQ are similar by the AA (Angle-Angle) similarity criterion:
- ∠OMP ∠OMQ as established
- ∠OMO ∠OMO (common angle)
- This means the ratio of the corresponding sides of the triangles is equal, i.e., PM/OM MQ/OM.
Proportional Segments
- From the similarity of triangles, we have:
- PM/OM MQ/OM
- Multiplying both sides by OM (which is non-zero), we get:
- PM MQ
Conclusion
Thus, we have shown that PM MQ, proving that the segments from the midpoint M to points P and Q are equal as required.
Visualizing the Butterfly Theorem
This theorem is often understood and visualized by looking at a picture of a circle with chords and the midpoint, showcasing the symmetry and balance involved.