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Point relation in a square (Posted on 2024-06-20) Difficulty: 3 of 5
In square ABCD, we select a point P such that AP=a, BP=2a, CP=3a. Find ∠APB?

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution Solution and Desmos chart Comment 3 of 3 |
Let ABCD be (0,0), (1,0), (1,1), (0,1)
The locus of points that is a certain ratio from 2 points is a circle.

A and B:
2 points that satisfy are (1/3,0) and (-1,0).  The circle has 
center (-1/3,0)
radius 2/3
Equation:  (x + 1/3)^2 + y^2 = 4/9


A and C:
2 points that satisfy are (1/4, 1/4) and (-1/2, -1/2).  The circle has 
center (-1/8, -1/8)
radius 3√2/8
Equation:  (x + 1/8)^2 + (y + 1/8)^2 = 9/32

With a little help from Wolfram Alpha, the 2 equations intersect at

x = 1/17 + 3√2/17 = 0.3083906286540756
y = -4/17 + 5√2/17  = 0.1806510477567927

BAP is 180*atan(y/x)/pi = 30.36119340482172 degrees
ABP is 180*atan(y/(1-x))/pi = 14.63880659517829 degrees
APB is 180 - ABP - BAP = 135 degrees

https://www.desmos.com/calculator/6flagrjlgu

  Posted by Larry on 2024-06-21 12:55:45
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