Geometry Probem

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Maybe I'm missing something but it seems simpler to go with ratios and defi ne R'=R + D in your diagram. Then by law of sines R'/R= W/X, and then H is re-written as H= (R'-R) + (R-X )= R'-X or R'= H+X, which is one d own, and then R=X/W*R'=X/W*(H+X). Is that really easy, or did I make a dumb mistake?

By golly, I think you've got it ;-) ...Jim Thompson

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fine R'=R + D in your diagram. Then by law of sines R'/R= W/X, and then H is re-written as H= (R'-R) + (R-X )= R'-X or R'= H+X, which is one down, and then R=X/W*R'=X/W*(H+X). Is that really easy, or did I make a dumb mistake?

I misread X for your D+R-H=R'-H, so it was a dumb mistake.

But it still boils down to (W/X*R-H)^2+(X/2)^2=R^2 for the right triangle with R, X/2, and R'-H sides. And this is a simple quadratic in R:

R= [ 2*W/X*H +/- SQRT((2*W/X*H)^2-4*[(W/X)^2-1]*(H^2+(X/2)^2))]/(2*[(W/X) ^2-1]) if you absolutely have to have closed form.

Yeah, there was a dumb error in mine.

The correct answer is

R = (X^2*(4*H^2 - W^2 + X^2)^(1/2) + 2*H*W*X)/(2*W^2 - 2*X^2) or -(X^2*(4*H^2 - W^2 + X^2)^(1/2) - 2*H*W*X)/(2*W^2 - 2*X^2) D = R * (W/X -1)

which is probably the same as yours

It's not immediately obvious to me how you pick the correct answer, both are positive and look 'reasonable' in the example I checked.

X = 22.36068 W = 38.78512 H = 16

R = 15.0000 (correct) or 12.6718 (incorrect)

corresponding to

D = 11.0000 (correct) or 9.2927 (incorrect)

Best regards, Spehro Pefhany

Not quite, see my follow-up post to RP's.

Best regards, Spehro Pefhany