Inherent Triangle Similarity

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 { ormalsize $cleft({c_{2}}-{c_{1}} ight)=bleft({b_{2}}-{b_{1}} ight)-aleft({a_{2}}-{a_{1}} ight)$. This latter equation can be written as { ormalsize ${c_{2}^{{2}}}-{c_{1}^{{2}}}=({b_{2}^{{2}}}-{b_{1}^{{2}}})-({a_{2}^{{2}}}-{a_{1}^{{2}}})$ 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.