# orderedItems[].content
A problem mentioned elsewhere; what is the largest similar triangle that can be embedded into a triangle?
i.e 3:4:5.
For any triangle, we can determine a lower bound (in fact I think it is sharp).
Take \(x,y,z\) as the vertices and \(X,Y,Z\) as the opposite sides and
\( x_{\theta},y_{\theta},z_{\theta}\) as the corresponding angles.
Example area of the triangle: \( A_{z}=\frac{1}{2}\cdot X\cdot Y\cdot\sin\left(z_{\theta}\right)\)
But \( A=A_{z}=A_{y}=A_{x}\)
The same for any other vertice order.
Join the midpoints of the sides to form vertices \( x', y', z'\) lying on the sides \( X, Y, Z\ )\)
And \(𝑋',𝑌',𝑍' \) the opposite sides of the new triangle.
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 \(X, Y, Z\).
Taking the example, we have
\(A_{z}'=\frac{1}{2}\frac{X}{2}\cdot\frac{Y}{2}\cdot\sin\left(z_{\theta}\right)=\frac{1}{4}\cdot\left(\frac{1}{2}\cdot X\cdot Y\cdot\sin\left(z_{\theta}\right)\right) \)
and the same for the other three vertices since the angles haven't changed.
Similarly for \(x_{\theta},y_{\theta}\)
So the area of the inside triangle is:
\(A-A'_{z}-A'_{x}-A'_{y}=\frac{1}{4}A \)
So we have that \(\frac{1}{4}\) works but is it maximum?
I am going to read
Perpendicular Polygons
https://www. jstor.org/stable/2322190
Which looks to be a paywall, but is available for reading (after numerous clicks).
Courtesy of our ongoing squabbles about “Intellectual Property” (as if it was a rock in my yard)
In any case, this starts out as sort of, Freshman Complex Analytic Geometry and rapidly gets into material that requires thinking

# orderedItems[].contentMap.en
A problem mentioned elsewhere; what is the largest similar triangle that can be embedded into a triangle?
i.e 3:4:5.
For any triangle, we can determine a lower bound (in fact I think it is sharp).
Take \(x,y,z\) as the vertices and \(X,Y,Z\) as the opposite sides and
\( x_{\theta},y_{\theta},z_{\theta}\) as the corresponding angles.
Example area of the triangle: \( A_{z}=\frac{1}{2}\cdot X\cdot Y\cdot\sin\left(z_{\theta}\right)\)
But \( A=A_{z}=A_{y}=A_{x}\)
The same for any other vertice order.
Join the midpoints of the sides to form vertices \( x', y', z'\) lying on the sides \( X, Y, Z\ )\)
And \(𝑋',𝑌',𝑍' \) the opposite sides of the new triangle.
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 \(X, Y, Z\).
Taking the example, we have
\(A_{z}'=\frac{1}{2}\frac{X}{2}\cdot\frac{Y}{2}\cdot\sin\left(z_{\theta}\right)=\frac{1}{4}\cdot\left(\frac{1}{2}\cdot X\cdot Y\cdot\sin\left(z_{\theta}\right)\right) \)
and the same for the other three vertices since the angles haven't changed.
Similarly for \(x_{\theta},y_{\theta}\)
So the area of the inside triangle is:
\(A-A'_{z}-A'_{x}-A'_{y}=\frac{1}{4}A \)
So we have that \(\frac{1}{4}\) works but is it maximum?
I am going to read
Perpendicular Polygons
https://www. jstor.org/stable/2322190
Which looks to be a paywall, but is available for reading (after numerous clicks).
Courtesy of our ongoing squabbles about “Intellectual Property” (as if it was a rock in my yard)
In any case, this starts out as sort of, Freshman Complex Analytic Geometry and rapidly gets into material that requires thinking

# orderedItems[].content
Introduction to an exposition on some work I have done over the last decade. Since I believe that "motivation" in mathematics is the best motivator 😀 . Explanations, Theorems, etc ... will follow; slowly I am reviewing the documentation (for the 11th time) again.
1) In the realm of polynomial generating functions, EGF's (Sheffer sequences https://en.wikipedia.org/wiki/Sheffer_ sequence), OGF's, and similar sequences are all "similar" when expressed as polynomial coefficient arrays/matrices.
#EGF_OGF
1/n

# orderedItems[].contentMap.en
Introduction to an exposition on some work I have done over the last decade. Since I believe that "motivation" in mathematics is the best motivator 😀 . Explanations, Theorems, etc ... will follow; slowly I am reviewing the documentation (for the 11th time) again.
1) In the realm of polynomial generating functions, EGF's (Sheffer sequences https://en.wikipedia.org/wiki/Sheffer_ sequence), OGF's, and similar sequences are all "similar" when expressed as polynomial coefficient arrays/matrices.
#EGF_OGF
1/n

# orderedItems[].content
#EGF_OGF
Of course, \( [P(r)] \) could have been applied to the left. Both change the polynomial value. But we can change the basis and keep the polynomial value.
\[ \left[\begin{array}{ccccc}
a_{0} & a_{1} & a_{2} & \cdots & a_{n}\end{array}\right]\left[\begin{array}{c}
P(r)\end{array}\right]^{-1}\left[\begin{array}{c}
P(r)\end{array}\right]\left[\begin{array}{c}
1\\
x\\
x^{2}\\
\cdots\\
x^{n}
\end{array}\right] \]
4/n :)

# orderedItems[].contentMap.en
#EGF_OGF
Of course, \( [P(r)] \) could have been applied to the left. Both change the polynomial value. But we can change the basis and keep the polynomial value.
\[ \left[\begin{array}{ccccc}
a_{0} & a_{1} & a_{2} & \cdots & a_{n}\end{array}\right]\left[\begin{array}{c}
P(r)\end{array}\right]^{-1}\left[\begin{array}{c}
P(r)\end{array}\right]\left[\begin{array}{c}
1\\
x\\
x^{2}\\
\cdots\\
x^{n}
\end{array}\right] \]
4/n :)

# orderedItems[].content
@rrogers Just Latex practice:
Utility of Pascal's Matrix
\[ \left[\begin{array}{ccccc}
a_{0} & a_{1} & a_{2} & \cdots & a_{n}\end{array}\right]\left[\begin{array}{c}
P(r)\end{array}\right]\left[\begin{array}{c}
1\\
x\\
x^{2}\\
\cdots\\
x^{n}
\end{array}\right] \]
\[ =\left[\begin{array}{ccccc}
a_{0} & a_{1} & a_{2} & \cdots & a_{n}\end{array}\right]\left[\begin{array}{c}
1\\
\left(x+r\right)\\
\left(x+r\right)^{2}\\
\cdots\\
\left(x+r\right)^{n}
\end{array}\right] \]
#EGF_OGF
3/n

# orderedItems[].contentMap.en
@rrogers Just Latex practice:
Utility of Pascal's Matrix
\[ \left[\begin{array}{ccccc}
a_{0} & a_{1} & a_{2} & \cdots & a_{n}\end{array}\right]\left[\begin{array}{c}
P(r)\end{array}\right]\left[\begin{array}{c}
1\\
x\\
x^{2}\\
\cdots\\
x^{n}
\end{array}\right] \]
\[ =\left[\begin{array}{ccccc}
a_{0} & a_{1} & a_{2} & \cdots & a_{n}\end{array}\right]\left[\begin{array}{c}
1\\
\left(x+r\right)\\
\left(x+r\right)^{2}\\
\cdots\\
\left(x+r\right)^{n}
\end{array}\right] \]
#EGF_OGF
3/n