New Discoveries About Triangles

New Discoveries About Triangles

The book 'New Discoveries About Triangles' includes two new theorems discovered by George Uthup, the author.

(1)Uthup's altitude theorem  and (2) Uthup's General theorem 

Both TRUE for arbitrary Triangles (any type of).

Uthup's altitude theorem gives a new method for drawing the altitudes of a triangle by drawing circles on the sides of the triangle with side lengths as diameters. Circles drawn on any two sides intersect each other at a point on the third side and the line joining this point to the opposite vertex will be the altitude to the third side. Similarly, all three altitudes can be drawn.

Uthup's General theorem is a generalization of the famous Pythagorean theorem which is  TRUE only in right-angled Triangles. The Pythagorean formula is a2+b2=c2 where a, and b are the sides containing the right angle and c the hypotenuse. But this General theorem is TRUE  for any type of Triangles.  If the side lengths of the triangle are a,b,c, and h the altitude to the side c which divides the side length c into two line segments of length m,n then the formula of this theorem is a2+b2=c2+u2  where u2 =2(h2-mn). Here we need not measure the lengths of m ,n, and h. Their values can be calculated by the formula in the book using only the side lengths of the triangle.  Also if the triangle is right-angled then u2  becomes zero so that the equation reduces to the Pythagorean formula a2+b2=c2.  Proofs of these two theorems are given in the book.

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