Proof 3.7c
Proof: Consider the right with a right angle at (Definition 21). Construct similar triangles on the sides of. Notice, by construction, and . We know that the area of a triangle can be found using the formula, . To find the area of , we would use the length of times length of times . To find the area of , we would use the length of times times . To find the area of , we would use the length of times length of times . Since is congruent to , we know that the areas must also be equal by Common Notion 1. Since is composed of two smaller triangles, we can their two ares together to find the whole area of the triangle. To do this, we would add the areas of and . We know by construction, , so this value can be substituted. In this case, the area is because of similar triangles as discussed with Ceva's Theorem and the addition properties previously discussed. If we complete the algebra, as follows:
(if we multiply everything by )
Therefore, we can conclude, by using similar triangles, that when a and b are legs of a right triangle and c is the hypotenuse.