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Two ladders (Posted on 2013-03-12) Difficulty: 3 of 5
Given two crossed ladders resting against two parallel walls.
The feet of the ladders are each at the base of the opposite wall.

Denote the height at which they cross by h and the lengths of the ladders by m and n.

1. Find the heights (h1,h2) at which the ladders touch the walls.
2. Show that 2*h is the harmonic average of h1 and h2.
3. Evaluate the distance (D) between the walls.
4. Provide an example such that m,n,h,h1,h2 and D are integers.

No Solution Yet Submitted by Ady TZIDON    
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Some Thoughts Thoughts Comment 1 of 1
1.  These follow from the Pythagorean Theorem. h1=sqrt(m^2-D^2) and h2=sqrt(n^2-D^2)

2. A very interesting finding, which I suspect is generalizable to all trapezoids. (No proof yet)

3. Part 2 implies that the ratio of h1:h2:h is independent of D.  That fact I have seen used in another puzzle (but I cannot seem to find it.)

4. I assume you want something nontrivial: m!=n.  Then I have m=119, n=70, h=30, h1=105, h2=42, D=56

Edited on January 1, 2017, 9:19 am
  Posted by Brian Smith on 2017-01-01 00:32:22

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