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"content": "<p>A problem mentioned elsewhere; what is the largest similar triangle that can be embedded into a triangle?<br /> i.e 3:4:5.<br />For any triangle, we can determine a lower bound (in fact I think it is sharp).<br />Take \\(x,y,z\\) as the vertices and \\(X,Y,Z\\) as the opposite sides and<br />\\( x_{\\theta},y_{\\theta},z_{\\theta}\\) as the corresponding angles.<br />Example area of the triangle: \\( A_{z}=\\frac{1}{2}\\cdot X\\cdot Y\\cdot\\sin\\left(z_{\\theta}\\right)\\)<br />But \\( A=A_{z}=A_{y}=A_{x}\\)<br />The same for any other vertice order.<br />Join the midpoints of the sides to form vertices \\( x', y', z'\\) lying on the sides \\( X, Y, Z\\ )\\) <br />And \\(𝑋',𝑌',𝑍' \\) the opposite sides of the new triangle.<br />Trim the “outside” triangles; remembering that one only needs to specify 3 parameters to determine the triangle and “similarity” is just a scaling of terms \\(X, Y, Z\\).<br />Taking the example, we have<br />\\(A_{z}'=\\frac{1}{2}\\frac{X}{2}\\cdot\\frac{Y}{2}\\cdot\\sin\\left(z_{\\theta}\\right)=\\frac{1}{4}\\cdot\\left(\\frac{1}{2}\\cdot X\\cdot Y\\cdot\\sin\\left(z_{\\theta}\\right)\\right) \\) <br />and the same for the other three vertices since the angles haven't changed.<br />Similarly for \\(x_{\\theta},y_{\\theta}\\)<br />So the area of the inside triangle is: <br />\\(A-A'_{z}-A'_{x}-A'_{y}=\\frac{1}{4}A \\) <br />So we have that \\(\\frac{1}{4}\\) works but is it maximum? <br />I am going to read<br />Perpendicular Polygons<br /><a href=\"https://www.jstor.org/stable/2322190\" target=\"_blank\" rel=\"nofollow noopener noreferrer\" translate=\"no\"><span class=\"invisible\">https://www.</span><span class=\"\">jstor.org/stable/2322190</span><span class=\"invisible\"></span></a><br />Which looks to be a paywall, but is available for reading (after numerous clicks). <br />Courtesy of our ongoing squabbles about “Intellectual Property” (as if it was a rock in my yard) <br />In any case, this starts out as sort of, Freshman Complex Analytic Geometry and rapidly gets into material that requires thinking</p>",
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"en": "<p>A problem mentioned elsewhere; what is the largest similar triangle that can be embedded into a triangle?<br /> i.e 3:4:5.<br />For any triangle, we can determine a lower bound (in fact I think it is sharp).<br />Take \\(x,y,z\\) as the vertices and \\(X,Y,Z\\) as the opposite sides and<br />\\( x_{\\theta},y_{\\theta},z_{\\theta}\\) as the corresponding angles.<br />Example area of the triangle: \\( A_{z}=\\frac{1}{2}\\cdot X\\cdot Y\\cdot\\sin\\left(z_{\\theta}\\right)\\)<br />But \\( A=A_{z}=A_{y}=A_{x}\\)<br />The same for any other vertice order.<br />Join the midpoints of the sides to form vertices \\( x', y', z'\\) lying on the sides \\( X, Y, Z\\ )\\) <br />And \\(𝑋',𝑌',𝑍' \\) the opposite sides of the new triangle.<br />Trim the “outside” triangles; remembering that one only needs to specify 3 parameters to determine the triangle and “similarity” is just a scaling of terms \\(X, Y, Z\\).<br />Taking the example, we have<br />\\(A_{z}'=\\frac{1}{2}\\frac{X}{2}\\cdot\\frac{Y}{2}\\cdot\\sin\\left(z_{\\theta}\\right)=\\frac{1}{4}\\cdot\\left(\\frac{1}{2}\\cdot X\\cdot Y\\cdot\\sin\\left(z_{\\theta}\\right)\\right) \\) <br />and the same for the other three vertices since the angles haven't changed.<br />Similarly for \\(x_{\\theta},y_{\\theta}\\)<br />So the area of the inside triangle is: <br />\\(A-A'_{z}-A'_{x}-A'_{y}=\\frac{1}{4}A \\) <br />So we have that \\(\\frac{1}{4}\\) works but is it maximum? <br />I am going to read<br />Perpendicular Polygons<br /><a href=\"https://www.jstor.org/stable/2322190\" target=\"_blank\" rel=\"nofollow noopener noreferrer\" translate=\"no\"><span class=\"invisible\">https://www.</span><span class=\"\">jstor.org/stable/2322190</span><span class=\"invisible\"></span></a><br />Which looks to be a paywall, but is available for reading (after numerous clicks). <br />Courtesy of our ongoing squabbles about “Intellectual Property” (as if it was a rock in my yard) <br />In any case, this starts out as sort of, Freshman Complex Analytic Geometry and rapidly gets into material that requires thinking</p>"
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