Dear colleagues and enthusiasts of beautiful geometric problems, I invite you to solve another elegant problem:
Reconstruct triangle ABC from points A, D, E, and H1.
I will be glad to see your solutions and share my own!
Peter Breuer
The method you have proposed for solving this is an algebraic approach and would be valid only when AH/EH = ... (see the picture).
On the other hand, I suggest resolving this geometrically.
Peter Breuer
There is an inaccuracy in your formula, and it will only be valid under a specific ratio as shown in the photo below. In all other cases, unfortunately, it is incorrect.
Peter Breuer
You are looking for the ratio AH/EH – that's correct!
There was an error in your formula; the correct formula is as follows:
2/(1-tg^2(ABC/2)).
You need to use the formula for the tangent of a double angle:
tg(ABC) = (2tg(ABC/2))/(1-tg^2(ABC/2))
Peter Breuer
Now your response is perfect!
Thank you for your interest and perseverance!
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