Hey genius minds, this is Swaroop Dora welcoming you to Mind Your Concept. I found a nice Theorem of chords of a circle- New Chord Progressive Theorem New Chord Progressive Theorem Theorem : If in a circle 3 Chord origin from a common point and angle between them is given. Then the following relation holds Zsin(α+ß)=Xsinß+Ysinα New Chord Progressive Theorem proof Spacial Case For Chord Progressive Theorem If α and ß are given same then Z=(x+y)/(2cosα). Conclusion Hope you like this Theorem. Thank you for reading my new work. Share it with your friends and family. Important links Visit my Instagram page for fun and interesting math puzzles by Swaroop Dora @MindYourConcept This Theorem I have used in my proof Ptolemy's Theorem
Power of Chord Theorem
Hey all, this is Swaroop Dora, welcoming you to Mind Your Concept. I have found a beautiful formula - Power of Chord Theorem. I have provided proof in my way, you can send your proof to me too.
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| Power of Chord Theorem - Mind Your Concept |
THEOREM STATEMENT: If in a circle two intersecting chords and angle between them are given the radius of the circle is guided by the following formula-
4R²sin²θ=(x-y)²+(w-z)²+4xysin²θ+2(x-y)(w-z)cosθ
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Proof of Power of Chord Theorem:
In our given figure draw line l and m perpendicular to AB and CD.
Now we find a quadrilateral OPRQ. We will now find the length of sides of quadrilateral OPRQ.
LET separate the quadrilateral from our circle to have a better imagination. In this quadrilateral OPRQ we know all side lengths (although we do need all side lengths in our calculations). Rather using any existing formula we will have some construction. DRAW RS and ST perpendicular to RQ and OQ. Now we get a rectangle SRQT. ∠SOT=θ, as exterior angle of a cyclic quadrilateral is equal to interior opposite angle. ∠ROP=θ, as corresponding angles in two parallel lines are equal.
Now simplify the final equation...
Now we get our formula derived. That is 4R²sin²θ=(x-y)²+(w-z)²+4xysin²θ+2(x-y)(w-z)cosθ. On further simplifying we are left with 4R²sin²θ=x²+y²+z²+w²-4xycos²θ+2(x-y)(w-z)cosθ.
General case:
When angle θ=90 degrees.
You can too find generalizations for angles- 30°, 45°, 60°. I left it as homework for you in comment.
Conclusion
Hope you like this Power of Chord Theorem. If you have any further thoughts on this Power of Chord Theorem, share with me in comment. If you have any doubts about the Theorem, comment.
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