According to Pythagoreans theorem these are an unexplained properties of all triangles without using the Law of Sine and Cosine.
\(\displaystyle (c\times\sin A)^2+(a\times\cos C)^2=a^2\)
\(\displaystyle (c\times\sin B)^2+(b\times\cos C)^2=b^2\)
\(\displaystyle (a\times\cos A)^2+(b\times\cos B)^2=c^2\)
\(\displaystyle (b\times\cos C)+(c\times\cos B)=a\)
\(\displaystyle (a\times\cos C)+(c\times\cos A)=b\)
\(\displaystyle (a\times\cos B)+(b \times\cos A)=c\)
\(\displaystyle (c\times\cos B)+(b \times\cos C)=a\)
\(\displaystyle (c\times\cos A)+(a\times\cos C)=b\)
and to find three of the altitudes given three sides:
\(\displaystyle h_a\frac{\sqrt{2(a^2 b^2+a^2 c^2+b^2 c^2)-a^4-b^4-c^4}}{2a}\)
\(\displaystyle h_b\frac{\sqrt{2(a^2 b^2+a^2 c^2+b^2 c^2)-a^4-b^4-c^4}}{2b}\)
\(\displaystyle h_c\frac{\sqrt{2(a^2 b^2+a^2 c^2+b^2 c^2)-a^4-b^4-c^4}}{2c}\)
I want to know if it is new ?
\(\displaystyle (c\times\sin A)^2+(a\times\cos C)^2=a^2\)
\(\displaystyle (c\times\sin B)^2+(b\times\cos C)^2=b^2\)
\(\displaystyle (a\times\cos A)^2+(b\times\cos B)^2=c^2\)
\(\displaystyle (b\times\cos C)+(c\times\cos B)=a\)
\(\displaystyle (a\times\cos C)+(c\times\cos A)=b\)
\(\displaystyle (a\times\cos B)+(b \times\cos A)=c\)
\(\displaystyle (c\times\cos B)+(b \times\cos C)=a\)
\(\displaystyle (c\times\cos A)+(a\times\cos C)=b\)
and to find three of the altitudes given three sides:
\(\displaystyle h_a\frac{\sqrt{2(a^2 b^2+a^2 c^2+b^2 c^2)-a^4-b^4-c^4}}{2a}\)
\(\displaystyle h_b\frac{\sqrt{2(a^2 b^2+a^2 c^2+b^2 c^2)-a^4-b^4-c^4}}{2b}\)
\(\displaystyle h_c\frac{\sqrt{2(a^2 b^2+a^2 c^2+b^2 c^2)-a^4-b^4-c^4}}{2c}\)
I want to know if it is new ?
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