Question 1182471 and its unexpected Links to Optics and Electronics (not forgetting crossed Ladders)
Sunday 24 May 2026 at 00:12
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Edited by Richard Walker, Sunday 24 May 2026 at 00:48
A few days back I posted a geometry problem, Question 1182471 from Math Stack Exchange, and then have a proof in a follow-up post. My proof was correct, I believe, and a decent effort , I thought.
But I must still have been thinking about it subconsciously, because yesterday I suddenly saw a much shorter and simpler solution, one that is also more insightful because it throws up an interesting link to optics, to electronics, and to the infamous Crossed Ladders Problem †
Here's the problem amd the (new) proof.
Now when I got to the equation at the end I realised I had seen it before! In optics it comes up in the Lens Formula, in electrical circuits it is the rule for the resistance formed by parallel resistors, and in the crossings ladders puzzle it is the height of the point where two ladders cross, expressed in terms of the heights the ladders reach up the walls of an alley.
The equation is also related to the Harmonic Mean, which has many uses, and to things like Egyptian fractions.
† Look it up. A problem that sees easy at first but isn't, it leads to 4thdegree equations.
Question 1182471 and its unexpected Links to Optics and Electronics (not forgetting crossed Ladders)
A few days back I posted a geometry problem, Question 1182471 from Math Stack Exchange, and then have a proof in a follow-up post. My proof was correct, I believe, and a decent effort , I thought.
But I must still have been thinking about it subconsciously, because yesterday I suddenly saw a much shorter and simpler solution, one that is also more insightful because it throws up an interesting link to optics, to electronics, and to the infamous Crossed Ladders Problem †
Here's the problem amd the (new) proof.
Now when I got to the equation at the end I realised I had seen it before! In optics it comes up in the Lens Formula, in electrical circuits it is the rule for the resistance formed by parallel resistors, and in the crossings ladders puzzle it is the height of the point where two ladders cross, expressed in terms of the heights the ladders reach up the walls of an alley.
The equation is also related to the Harmonic Mean, which has many uses, and to things like Egyptian fractions.
† Look it up. A problem that sees easy at first but isn't, it leads to 4th degree equations.