Abstract
This paper shows many relationships for a triangle by using its altitudes to form inner triangles that have a three 4-fold similarity. The altitudes partition the sides of the triangle $a={a_{1}}+{a_{2}}, b={b_{1}}+{b_{2}}, c={c_{1}}+{c_{2}}$ into partial side lengths of ${a_{1}},{a_{2}},{b_{1}},{b_{2}},{c_{1}},{c_{2}}$. We show that ${a_{1}b_{2}c_{1}}={a_{2}b_{1}c_{2}}$ and {\normalsize $c\left({c_{2}}-{c_{1}}\right)=b\left({b_{2}}-{b_{1}}\right)-a\left({a_{2}}-{a_{1}}\right)$. This latter equation can be written as {\normalsize ${c_{2}^{{2}}}-{c_{1}^{{2}}}=({b_{2}^{{2}}}-{b_{1}^{{2}}})-({a_{2}^{{2}}}-{a_{1}^{{2}}})$or }{${a_{1}^{{2}}}+{b_{2}^{{2}}}+{c_{1}^{{2}}}={a_{2}^{{2}}}+{b_{1}^{{2}}}+{c_{2}^{{2}}}$}.We also note that ${h_{1}h_{2}}={h_{3}h_{4}}={h_{5}h_{6}}$,where ${h_{1}}+{h_{2}},{h_{3}}+{h_{4}},{h_{5}}+{h_{6}}$are the altitudes of the triangle. These concise relationships for a triangle are based on its inherent similarity, and provide for simple equations, similar to the Pythagorean Theorem for right triangles.
Highlights
In mathematics as in life, many relationships are established from a simple observation
We show that a1b2c1 = a2b1c2 and c (c2 − c1) = b (b2 − b1) − a (a2 − a1)
We note that h1h2 = h3h4 = h5h6, where h1 + h2, h3 + h4, h5 + h6 are the altitudes of the triangle
Summary
In mathematics as in life, many relationships are established from a simple observation. We use a simple geometrical observation to show a three 4-fold similarity of triangles. These triangles are formed using the altitudes of a triangle. We use this similarity to establish a list of many relationships for a triangle. From the fact that corresponding angles of similar triangles are similar:. These angles are shown in Fig.: As evident from Fig. (and the previous equations): Theorem 1. Similar right triangles are formed giving the three 4-fold similarity of triangles:
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