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The mathematician formerly known as "Day Late Don".

Honk if you love the Function Monkey.

See some of my research (and fun stuff) at my blog, "The Bloog!"


Dec
17
comment Construct quadrangle with given angles and perpendicular diagonals
@HandeBruijn: BTW, using polys like $(a^2-bd)\sin\alpha - a(b+d)\cos\alpha$ (based on my (1), (2), (3), (4)) helps the process by reducing the system's overall degree. I also express the trigs as complex exponentials, to avoid sine-cosine conversions & other hassles; conveniently, the exponential expressions factor a bit. Once I've solved my quadratic for $a$ (or $a^2$), I restore the trigs & seek the soln's final form. The symbol-pushing challenge is to determine how best to factor $P+Q\sqrt{k}$ ---with unattractive $P$ & $Q$--- into $(P_1+Q_1\sqrt{k})(P_2+Q_2\sqrt{k})$, as in my $a/s$ above.
Dec
17
comment Construct quadrangle with given angles and perpendicular diagonals
@HandeBruijn: Hitting a big non-linear system like that with solve() is usually a bad idea; better to work in stages: hit the polynomial system $a^2+b^2+c^2+d^2-s^2$, $(a^2+d^2)(a^2+b^2)\cos^2\alpha-(a^2-bd)^2$, etc, w/Resultant[] (or equivalent), eliminating side-lengths one-by-one until a polynomial in (say) $a$ remains. Each normalization I listed yields a quadratic (in either $a$ or $a^2$), which is easily solved. The hard part is wrestling the soln's sprawling trig expressions into meaningful form. A Simplify[] sledgehammer isn't up to the task here; clever massage is needed.
Dec
16
revised Construct quadrangle with given angles and perpendicular diagonals
added 1264 characters in body
Dec
15
comment What do you call 'perpendicular but skew' lines?
"perpendicularly skew"? "lines with orthogonal direction vectors"? I'll note that an orthocentric tetrahedron (aka, an "orthogonal tetrahedron") is defined simply as "a tetrahedron where all three pairs of opposite edges are perpendicular"; of course, those opposite edges are skew, as well, and we implicitly agree that "perpendicular" in this context describes the directions.
Dec
15
comment Construct quadrangle with given angles and perpendicular diagonals
I've been poking at this stuff, looking for a better representation. Note that the ratio $b/d$ should have a comparable form to $c/a$, but it doesn't look that way from what's shown. This suggests to me that each of $a$, $b$, $c$, $d$ can/should be written in terms of a common core of some kind (as opposed to writing $b$, $c$, $d$ in terms of $a$). I've tried various normalizations, such as $a+b+c+d=s$, $a^2+b^2+c^2+d^2=s^2$, even $ab+bc+cd+da=s^2$ (which fixes the area of the quadrilateral). The resulting expressions are annoyingly complicated, however, so I haven't (yet!) found the key.
Dec
15
answered Chords of a circle
Dec
14
revised Relation between edgelengths in a tetrahedron with two right angles and three equal edges
added 25 characters in body
Dec
14
answered Relation between edgelengths in a tetrahedron with two right angles and three equal edges
Dec
14
revised prove that quadrilateral is cyclic
deleted 25 characters in body
Dec
13
revised Geometry Problem about tangent lines
edited body
Dec
13
answered Geometry Problem about tangent lines
Dec
13
comment Construct quadrangle with given angles and perpendicular diagonals
@HandeBruijn: Redundant, perhaps, but not irrelevant. When verifying my work, or seeking better representations for the values, or devising a construction, it's worth remembering that I could've written, say, $\cos(\alpha+\delta)$ for $\cos(\beta+\gamma)$, and so forth.
Dec
13
comment Trisecting an angle
You're welcome. And I used GeoGebra.
Dec
13
comment Trisecting an angle
+1 for questioning authority. :)
Dec
13
answered Trisecting an angle
Dec
13
answered prove that quadrilateral is cyclic
Dec
13
comment prove that quadrilateral is cyclic
Hint: The circle is tangent to $BC$ at $E$.
Dec
12
revised Construct quadrangle with given angles and perpendicular diagonals
added 1811 characters in body
Dec
12
answered Construct quadrangle with given angles and perpendicular diagonals
Dec
10
revised Ratio of sines given, find ratio of cosines?
added 5 characters in body