Camouflaged Butterfly

Source

Leo Giugiuc has kindly posted at the CutTheKnotMath facebook page a problem due to Miguel Ochoa Sanchez:

Camouflaged Butterfly - source

Problem

Chord $CD\;$ and $QT\;$ of a given circle meet at point $P.\;$ The tangents at $Q\;$ and $T\;$ cross $CD\;$ extended at $A\;$ and $B,\;$ respectively.

Camouflaged Butterfly - problem

Prove that $\displaystyle\frac{1}{AP}-\frac{1}{BP}=\frac{1}{CP}-\frac{1}{DP}.$

Solution 1

By the Power of a Point Theorem, $AC\cdot AD=AQ^2,\;$ $BD\cdot BC=BT^2,\;$ and $CP\cdot DP=PQ\cdot PT,\;$ implying $(AC(AP+DP)=AQ^2\;$ and $BD(BP+CP)=BT^2.\;$ By Stewart's theorem in $\Delta QAP\;$ for the cevian $QC,$

$CQ^2\cdot AP+AC\cdot AP\cdot CP=PQ^2\cdot AC+AQ^2\cdot CP,$

so that $CQ^2\cdot AP+AC\cdot AP\cdot CP=PQ^2\cdot AC+AC(AP+DP)\cdot CP,\quad$ or $CQ^2\cdot AP=PQ^2\cdot AC+AC\cdot CP\cdot DP,\;$ or else, $CQ^2\cdot AP=PQ^2\cdot AC+AC\cdot PQ\cdot PT,\;$ and, finally, $CQ^2\cdot AP=AC\cdot PQ\cdot PT.\;$ Similarly, $DT^2\cdot BP=BD\cdot PT\cdot QT.$

We thus have $\displaystyle\frac{QC^2}{DT^2}=\frac{BP\cdot AC\cdot PQ}{AP\cdot BD\cdot PT}.\;$ But triangles $PCQ\;$ and $PTD\;$ are similar, so that $\displaystyle\frac{QC^2}{DT^2}=\frac{PQ^2}{DP^2},\;$ or $\displaystyle\frac{PQ^2}{DP^2}=\frac{BP\cdot AC\cdot PQ}{AP\cdot BD\cdot PT},\;$ implying $\displaystyle\frac{1}{DP^2}=\frac{BP\cdot AC}{AP\cdot BD\cdot PT\cdot PQ},\;$ or $\displaystyle\frac{1}{DP^2}=\frac{BP\cdot AC}{AP\cdot BD\cdot CP\cdot DP}.\;$ Simplifying, $\displaystyle\frac{AP\cdot CP}{BP\cdot DP}=\frac{AC}{BD},\;$ i.e., $\displaystyle\frac{AP\cdot CP}{BP\cdot DP}=\frac{AP-CP}{BP-DP},\;$ which is $\displaystyle\frac{BP-DP}{BP\cdot DP}=\frac{AP-CP}{AP\cdot CP},\;$ or $\displaystyle\frac{1}{DP}-\frac{1}{BP}=\frac{1}{CP}-\frac{1}{AP},\;$ as required.

Solution 2

Draw $AQ\parallel BS.\;$ We'll use the notations as defined in the diagram below:

Camouflaged Butterfly - solution 2

We have $\displaystyle\frac{a}{b}=\frac{AO}{BS}=\frac{AQ}{BT}.\;$ From this

$\displaystyle\left(\frac{a}{b}\right)^2=\left(\frac{AQ}{BT}\right)^2=\frac{AD\cdot AC}{BC\cdot BD}=\frac{(a+n)(a-m)}{(b+m)(b-n)},$

From this we deduce

$\displaystyle\frac{(a+n)(a-m)}{a^2}=\frac{(b+m)(b-n)}{b^2},$

which is the same as $\displaystyle\frac{1}{a}-\frac{1}{b}=\frac{1}{m}-\frac{1}{n},\;$ as required.

Remark

The problem above is a clear generalization of the one, Butterfly in Inscriptible Quadrilateral. Furthermore, the genuine Butterfly Theorem in which the fact that $P\;$ is the midpoint of one of the segments, $AB,\;$ $CD,\;$ implies its being the midpoint of the other, has been generalized to exactly same condition $\displaystyle\frac{1}{DP}-\frac{1}{BP}=\frac{1}{CP}-\frac{1}{AP},\;$ for an arbitrary $P,\;$ see, e.g., the remark at the end of Proof 8 of the Butterfly Theorem.

Acknowledgment

Solution 1 is by Leo Giugiuc and Dan Sitaru; Solution 2 is by Miguel Ochoa Sanchez.

 

Butterfly Theorem and Variants

  1. Butterfly theorem
  2. 2N-Wing Butterfly Theorem
  3. Better Butterfly Theorem
  4. Butterflies in Ellipse
  5. Butterflies in Hyperbola
  6. Butterflies in Quadrilaterals and Elsewhere
  7. Pinning Butterfly on Radical Axes
  8. Shearing Butterflies in Quadrilaterals
  9. The Plain Butterfly Theorem
  10. Two Butterflies Theorem
  11. Two Butterflies Theorem II
  12. Two Butterflies Theorem III
  13. Algebraic proof of the theorem of butterflies in quadrilaterals
  14. William Wallace's Proof of the Butterfly Theorem
  15. Butterfly theorem, a Projective Proof
  16. Areal Butterflies
  17. Butterflies in Similar Co-axial Conics
  18. Butterfly Trigonometry
  19. Butterfly in Kite
  20. Butterfly with Menelaus
  21. William Wallace's 1803 Statement of the Butterfly Theorem
  22. Butterfly in Inscriptible Quadrilateral
  23. Camouflaged Butterfly
  24. General Butterfly in Pictures
  25. Butterfly via Ceva
  26. Butterfly via the Scale Factor of the Wings
  27. Butterfly by Midline
  28. Stathis Koutras' Butterfly
  29. The Lepidoptera of the Circles
  30. The Lepidoptera of the Quadrilateral
  31. The Lepidoptera of the Quadrilateral II
  32. The Lepidoptera of the Triangle
  33. Two Butterflies Theorem as a Porism of Cyclic Quadrilaterals
  34. Two Butterfly Theorems by Sidney Kung

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