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Constant Ratio (Posted on 2013-12-17) Difficulty: 2 of 5

  
A circle has diameter AB and a point P lies on AB between A and B.
A point X, distinct from A and B, lies on the circle.

Prove that tan(∠AXP)/tan(∠XAP) is constant for all values of X.
  

See The Solution Submitted by Bractals    
Rating: 4.0000 (1 votes)

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Solution Solution | Comment 1 of 2
Draw a line through A parallel to XB, and extend XP to cross this line at Q.

Triangles APQ and BPX are similar (AQ||XB giving alternate angles) (1)

Thus:       tan/AXP / tan/XAP       = |AX|tan/AXQ / |AX|tan/XAB

                                                = |AQ| / |BX|    using (1)

                                                = |AP| / |BP|   (a fixed ratio).



  Posted by Harry on 2013-12-17 17:17:20
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