Wikipedia:Reference desk/Archives/Mathematics/2013 November 4

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November 4

Irregular Quadrilateral

Not a home work question, just a curiosity.

Using

as a reference:

If angles CAD, BAC, and BCD and the lengths of sides b & c are known, is it possible solve sides a & d as well as line AC?

SRICE_{13 (TALK | EDITS)} 23:39, 4 November 2013 (UTC)

As drawn, sides a and d are equal. Is this meant, or should each lower-case letter be the length of the tangent from the upper-case letter? And is r given?86.176.135.38 (talk) 12:59, 5 November 2013 (UTC)

(edit conflict) Not at all a rigorous answer, but ... I would think so. If you know angle BCD, and sides b and c, you can draw sides BC and CD in Cartesian coordinates. If you know angles CAD and BAC, you know their ratio, so you can draw a curve emanating from C that maintains, for each point P on the curve, that constant ratio for CPD:BPC. As you extend the curve and P moves farther from C, the angles CPD and BPC can presumably be shown to decrease monotonically. So when these angles equal the required angles CAD and BAC, that location of point P gives you point A. So the framework should provide a uniqueness proof for the location of point A, which in turn gives the desired a, d, and AC. And maybe you could work it out exactly in Cartesian coordinates by letting the location of P be parametric on some parameter t; probably the function P(t) depends on some trigonometric functions. Duoduoduo (talk) 13:03, 5 November 2013 (UTC)