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 Nov18 comment Finding an angle between side and a segment from specified point inside an equilateral triangle It's simple: note that arc length = angle (in radians) when the radius of the circle is 1 (i.e. unit circle). So when the arc length is $u$, the chord length is $2\sin(u/2)$. Nov15 comment Find the line that intercepts the lines $r$ and $s$ and forms congruent angles to the coordinate axes You're almost done. From the first part of your argument, if you set $\overrightarrow{AB}=(x,\,y,\,z)$, you get $|x|=|y|=|z|$. (The angle is not 45 degrees by the way, but the argument still stands.) Then you have four cases to consider: $x=y=z$, $x=-y=z$, $x=y=-z$, $x=-y=-z$. Nov15 answered Nice geometry with areas Nov15 comment Finding an angle between side and a segment from specified point inside an equilateral triangle I used $\cos 3\theta=4\cos^3\theta-3\cos\theta$ and $\sin 2\theta=2\sin\theta\cos\theta$. So $\cos 3\theta=\sin 2\theta$ becomes $\cos\theta(4\cos^2\theta-3)=\cos\theta(2\sin\theta)$, and dividing by $\cos\theta$ (which is not zero) we get $4(1-\sin^2\theta)-3=2\sin\theta$. Nov14 comment Finding an angle between side and a segment from specified point inside an equilateral triangle A proof of $d_3+d_5=d_9$: it suffices to show $\sin 54^{\circ}-\sin 18^{\circ}=\sin 30^{\circ}$, that is $2\cos 36^{\circ}\sin 18^{\circ}=1/2$. Using the fact that $\alpha=\sin 18^{\circ}$ satisfies $4(1-\alpha^2)-3=2\alpha$ (from $\cos 54^{\circ}=\sin 36^{\circ}$), we can show that $2(1-2\alpha^2)\alpha=1/2$. Nov14 comment Finding an angle between side and a segment from specified point inside an equilateral triangle This lemma is stated in www-math.mit.edu/~poonen/papers/ngon.pdf (pages 4-5). I'll check if it is really true. Nov14 revised Finding an angle between side and a segment from specified point inside an equilateral triangle added 28 characters in body Nov14 answered Finding an angle between side and a segment from specified point inside an equilateral triangle Nov13 answered CD is height of right-angled triangle ABC, M and N are midpoints of CD and BD: prove AM⊥CN Nov13 revised How find this maximum of this $(1-x)(1-y)(10-8x)(10-8y)$ added 269 characters in body Nov11 awarded Promoter Nov11 revised Translational invariance and zero eigenvalue added 35 characters in body; edited tags Nov11 revised How find this maximum of this $(1-x)(1-y)(10-8x)(10-8y)$ deleted 171 characters in body Nov11 revised How find this maximum of this $(1-x)(1-y)(10-8x)(10-8y)$ deleted 1 character in body Nov11 revised How find this maximum of this $(1-x)(1-y)(10-8x)(10-8y)$ deleted 1 character in body Nov11 answered How find this maximum of this $(1-x)(1-y)(10-8x)(10-8y)$ Nov9 asked Translational invariance and zero eigenvalue Sep24 awarded Autobiographer Sep1 awarded Yearling Apr11 awarded Revival